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Russian Mathematical Surveys, 2024, Volume 79, Issue 4, Pages 567–648
DOI: https://doi.org/10.4213/rm10172e
(Mi rm10172)
 

This article is cited in 1 scientific paper (total in 1 paper)

Cohomology of Hopf algebras and Massey products

V. M. Buchstabera, F. Yu. Popelenskiibc

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Moscow Center of Fundamental and Applied Mathematics
c Faculty of Mathematics and Mechanics, Lomonosov Moscow State University
References:
Abstract: The theory of the trigraded Buchstaber spectral sequence $\operatorname{Bss}$ for graded Hopf algebras is developed. It is shown that the differentials of $\operatorname{Bss}$ define an increasing exhaustive filtration as a new structure in the cohomology of Hopf algebras. This structure is described explicitly for a number of known Hopf algebras.
For the tensor algebra $T(s \operatorname{Ext}^{1,*}_{A}(\Bbbk,\Bbbk))$ of the suspension of the one-dimensional cohomology of a Hopf algebra $A$ over a field $\Bbbk$, the construction of partial multivalued operations $\operatorname{Bss}_p$, $p\geqslant 1$, is presented. This construction is used to describe the differentials in the spectral sequence $\operatorname{Bss}$ and the exhaustive filtration in $\operatorname{Ext}_{A}^{*,*}(\Bbbk,\Bbbk)$.
It is shown that the structure introduced is an effective tool for solving several well-known problems: (1) realising cohomology classes of Hopf algebras by Massey products; (2) interpreting differentials in $\operatorname{Bss}$ as Massey operations; (3) effective construction of a certain class of Massey products in the form of differentials in $\operatorname{Bss}$.
Bibliography: 74 titles.
Keywords: Hopf algebras, Landweber–Novikov algebra, Buchstaber spectral sequence, Eilenberg–Moore spectral sequence, $\operatorname{Bss}$-operations, cohomology of nilmanifolds.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-15-2022-265
The work of V. M. Buchstaber was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-265).
Received: 14.03.2024
Bibliographic databases:
Document Type: Article
UDC: 512.66+515.14
Language: English
Original paper language: Russian

In memory of Sergei Petrovich Novikov

1. Introduction

Hopf algebras are fundamental objects of modern mathematics, which have important applications in mathematical physics. The theory and applications of Hopf algebras were the focus of attention of Sergei Novikov since his first publication [1] in 1959. The Landweber–Novikov algebra is one of the most famous Hopf algebras. The cohomology of this algebra is the initial term of the Adams–Novikov spectral sequence.

Apart from multiplication, there are many other important operations in the cohomology of Hopf algebras. In [1] and [2] the $\text{mod } p$ Steenrod operations in the cohomology of cocommutative Hopf algebras were introduced and found important applications.

The cohomology ring $H^*(\Omega X,\Bbbk)$ of the loop space $\Omega X$ with coefficients in the field $\Bbbk$ is a classical example of a Hopf algebra. The Milnor spectral sequence [3], [4] calculates the cohomology of $X$ from the cohomology of $\Omega X$ and leads to problems on the cohomology structure of the Hopf algebra $H^*(\Omega X,\Bbbk)$; see [5] for more details.

In [6], a spectral sequence was introduced for calculating cohomology of Hopf algebras over a field of finite characteristic.

The theory of quantum groups (see [7] and [8]) has led to new problems in the cohomology of Hopf algebras (over an arbitrary field), which arise, among other things, from the theory of deformations (see, for example, [9]).

For a number of results on modern problems in the cohomology of Hopf algebras, see the conference materials [10].

The Massey operations are defined in the cohomology of differential algebras (see [11]–[14]). These operations are often related to the differentials of appropriate spectral sequences (see [15]–[17]).

The focus of our work is the following problems:

(1) realisation of cohomology classes of Hopf algebras by Massey products;

(2) spectral sequences whose differentials are realised by Massey operations;

(3) effective constructions of a certain class of Massey products realisable as differentials of a spectral sequence.

We describe the results on problems (1)–(3) using an approach based on the Buchstaber spectral sequence $\operatorname{Bss}$. As examples, we consider two series of Hopf algebras that arise as the universal enveloping algebras of well-known Lie algebras:

$\bullet$ the Lie algebra $\mathcal{G}L^n$ of the group $L^n$ of polynomial transformations of the line;

$\bullet$ the Heisenberg–Lie algebra $\mathcal{GH}^{2n+1}$.

The cohomology of the Heisenberg–Lie algebras $\mathcal{GH}^{2n+1}$ is the subject of a number of papers; see, for example, [18]–[24]. Generalising the results inthese papers, we present a complete solution of problems (1), (2) and (3) in the case of the Lie algebras $\mathcal{GH}^{2n+1}_H$ (see § 8). Furthermore, we describe the cohomology ring of $\mathcal{GH}^{2n+1}$ as the module over the cohomology ring of the torus $T^{2n}$. Note that the nilmanifolds $M_H^{2n+1}$ admit contact structures. We pose the problem of the description of the manifold $M_H^{2n+1}$ as the total space of the bundle $M_H^{2n+1}\to V^n$ over an $n$-dimensional Abelian variety $V^n$.

The Buchstaber spectral sequence was introduced in [25] in connection with the algebraic problem of the computation of the second term $E_2(AN)$ of the Adams–Novikov spectral sequence for stable homotopy groups of spheres. Recall that

$$ \begin{equation*} E_2(AN)=\operatorname{Ext}_S^{s,t}(\mathbb{Z},\Omega_U^*),\qquad s\geqslant 0, \end{equation*} \notag $$
where $S$ is the Hopf algebra of stable Landweber–Novikov cohomology operations in complex cobordism $U^*(-)$, and $\Omega_U^*$ is the scalar ring of the theory $U^*(-)$. By the Milnor–Novikov theorem the ring $\Omega_U^*$ is isomorphic to the polynomial ring $\mathbb{Z}[y_1,y_2,\ldots]$, $|y_i|=-2i$ (see [26]–[29]). Here $|x|$ denotes the degree of an element $x$.

We extend the theory of the Buchstaber spectral sequence (denoted by $\operatorname{Bss}$) to the case of general Hopf algebras. As a result, we obtain a new structure in the cohomology ring of a Hopf algebra, which is determined by the differentials in $\operatorname{Bss}$. It is proved that these differentials define an increasing exhaustive filtration on the cohomology of the Hopf algebra and that, for a Hopf algebra over a field, any cohomology class in the $k$th filtration is a representative of a non-trivial $(k+1)$-fold iterated matric Massey product of one-dimensional classes. We introduce new $\operatorname{Bss}$-operations on the cohomology of Hopf algebras. These operations are defined in terms of the differentials in $\operatorname{Bss}$ and form a new structure on cohomology of Hopf algebras. A $\operatorname{Bss}$-operation is defined on the tensor algebra on the one-dimensional cohomology of a Hopf algebra, and takes values in the cohomology of this algebra. The $\operatorname{Bss}$-operations, like higher Massey products, are partially defined and multivalued. A $\operatorname{Bss}$-operation is said to be non-trivial if the set of its values does not contain zero. An important advantage of $\operatorname{Bss}$-operations over Massey products is that each cohomology class of a Hopf algebra over a field is realised by a unique non-trivial $\operatorname{Bss}$-operation, as shown in this paper.

As is known, Hopf algebras take their origin in the study of multiplications on topological spaces (see [30]). Other problems in algebraic topology and homological algebra related to spectral sequences have also had a major influence on the development of the theory of Hopf algebras.

In works by Drinfeld (see, for example, [7]) a new class of Hopf algebras was introduced under the name of ‘quantum groups’. They have the form $A\otimes A_*$ with deformed multiplication, where $A$ is a Hopf algebra and $A_*$ is the Hopf algebra dual to $A$. Quantum groups were used to solve the Yang–Baxter equation, well known in physics (see, for example, [8]). Works by Drinfeld and Jimbo formed the basis of the theory of quantum groups as a new field of research. Using the construction of quantum groups Novikov introduced the concept of doublings of Hopf algebras and, as a basic example, considered doublings of the Landweber–Novikov algebra [31]. Broad attention to these works has led to new directions of research and the construction of important examples of Hopf algebras in functional analysis, graph theory and enumerative combinatorics.

The original concept of the Buchstaber spectral sequence (see [25]) is as follows. Let $S_*$ denote the Hopf algebra that is dual over $\mathbb{Z}$ to the Landweber–Novikov algebra $S$. The algebra $S$ acts in the standard way on the ring $\Omega_U^*$. This action extends to $\Omega_U^*\otimes\mathbb{Q}$. In [25] the canonical inclusion $\Omega_U^*\subset \Omega_U^*\otimes \mathbb{Q}$ was decomposed into a composition $\Omega_U^*\subset\Omega_U^*(\mathbb{Z})\subset\Omega_U^*\otimes\mathbb{Q}$ of inclusions of $S$-modules. The ring $\Omega^U_*(\mathbb{Z})$ consists of elements of $\Omega^U_*\otimes \mathbb{Q}$ for which all Chern characteristic numbers are integers. The action of $S$ on $\Omega_U^*(\mathbb{Z})$ is determined by the action of $S$ on $\Omega_U^*\otimes \mathbb{Q}$. In [25] the remarkable isomorphism $\Omega_U^*(\mathbb{Z})=S_*$ was established, which is compatible with the action of $S$ on $\Omega_U^*(\mathbb{Z})$ and with the canonical action of $S$ on $S_*$.

In [25] and [32] a filtration of $\Omega_U^*(\mathbb{Z})$ by $S$-submodules was constructed. It starts from $\Omega_U^*\subset \Omega_U^*(\mathbb{Z})$, and its successive quotients are trivial $S$-modules:

$$ \begin{equation} \Omega_U^*=N_0\subset N_1\subset \cdots \subset \Omega_U^*(\mathbb{Z}). \end{equation} \tag{1} $$
This filtration defines a trigraded spectral sequence $E_r^{p,q,t}$ converging to $\operatorname{Ext}^{s,t}_S(\mathbb{Z},\Omega^*_U(\mathbb{Z}))=\operatorname{Ext}^{s,t}_S(\mathbb{Z},S_*)$. It is fundamentally important that $\operatorname{Ext}^{s,t}_S(\mathbb{Z},S_*)=0$ for $(s,t)\ne (0,0)$. Note that $\operatorname{Ext}^{0,0}_S(\mathbb{Z},S_*)=\mathbb{Z}$.

Now observe that

$$ \begin{equation*} E_1^{p,q,t}= \operatorname{Ext}_S^{-(p+q),t}(\mathbb{Z},{N_p}/{N_{p-1}}), \end{equation*} \notag $$
where $N_p$ are the filtration terms (1), and for $p>0$ the algebra $S$ acts trivially on the coefficient group $N_p/N_{p-1}$. Hence, for $p>0$ the group $E_1^{p,q,t}$ is expressed in terms of $\operatorname{Ext}_S^{*,*}(\mathbb{Z},\mathbb{Z})$ and $N_p/N_{p-1}$ using the short exact Künneth sequence.

On the other hand $E^{0,-s,t}_1=\operatorname{Ext}^{s,t}_S(\mathbb{Z},\Omega_U^*)$. As noted above, the Buchstaber spectral sequence converges to zero except for $E_\infty^{0,0,0}= \mathbb{Z}$, so the images of its differentials must exhaust $E^{0,*,*}_1= \operatorname{Ext}^{*,*}_S(\mathbb{Z},\Omega_U^*)$. This gives a new structure on $E^{0,-s,t}_1=\operatorname{Ext}^{s,t}_S(\mathbb{Z},\Omega_U^*)$, which connects the groups $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\Omega_U^*)$ recursively with the groups $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\mathbb{Z})$ and $N_p/N_{p-1}$. At the same time the algebraic mechanism connecting successive filtration quotients $N_p/N_{p-1}$ with the groups $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\Omega_U^*)$ and $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\mathbb{Z})$ has important topological applications (see [33]–[35]).

In §2.1 the results of [32]–[36] on the trigraded spectral sequence for $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\Omega_U^*)$ are described and a number of well-known topological applications are presented. A connection between the Landweber–Novikov algebra $S\otimes \mathbb{Q}$ and the universal enveloping of the Lie algebra $L_1$ of formal vector fields on the line that vanish together with the first derivative at the origin was discovered in [34]. This led to the calculation of the cohomology of the algebra $S\otimes \mathbb{Q}$ on the basis of Goncharova’s theorem on the cohomology of the algebra $L_1$ [37], [38]. In § 2.2 we state Goncharova’s theorem together with a corollary to it that led to Buchstaber’s conjecture on Massey products.

In § 2.3 and § 2.4 we discuss two important series of nilmanifolds $M^n$ and $M^{2k+1}_H$, which are the subject of §§ 7 and 8.

The first series belongs to the class of nilmanifolds introduced in [39]. There, the class of nilpotent Lie algebras of $V_n$ with basis $e_1,\dots,e_n$ and relations $[e_i,e_j]=(j-i)e_{i+j}$ for $i+j\leqslant n$, $[e_i,e_j]=0$ for $i+j> n$ was considered. By Malcev’s theorem the corresponding simply connected Lie group $V_n$ contains a family of cocompact lattices. In contrast to the commutative case, there are non-isomorphic lattices among them. According to Nomizu’s theorem, the real cohomology ring of any nilmanifold obtained as the quotient of $V_n$ by a cocompact lattice is naturally isomorphic to the cohomology ring of the Lie algebra $V_n$. However, the fundamental groups of these nilmanifolds can be non-isomorphic.

In [40] the group $L^n$ of polynomial transformations of the line was introduced. It is closely related to the Landweber–Novikov algebra. The group $L^n$ is a simply connected nilpotent Lie group and contains a lattice consisting of transformations with integer coefficients. The Lie algebra ${\mathcal G} L^n$ is isomorphic to $V_n$. The corresponding nilmanifold $M^n$ is the quotient of $L^n$ by the subgroup of transformations with integer coefficients; it was studied in [40].

The manifolds $M^n$ are related by a sequence of bundles

$$ \begin{equation*} \cdots \to M^{n+1}\to M^n\to \cdots \to M^3 \to M^2=T^2 \to M^1=S^1 \end{equation*} \notag $$
with fibre a circle.

Note that by Nomizu’s theorem (see [41]) the real cohomology of the nilmanifold $L^n/\Gamma$ does not depend on the particular choice of lattice $\Gamma$. Moreover, each compatible choice of lattices $\Gamma_n\subset L^n$ leads to the corresponding tower of bundles.

In § 8 we consider the $(2n+1)$-dimensional generalised Heisenberg group $\mathcal{H}^{2n+1}$ (the Heisenberg group $\mathcal{H}^{3}$ for $n=1$), realised as a special subgroup of $\mathrm{GL}_{n+2}(\mathbb{R})$. It is a simply connected nilpotent Lie group with an infinite family of lattices. For each $n$, we introduce the manifold $M_H^{2n+1}$ as the quotient of $\mathcal{H}^{2n+1}$ by the subgroup $\mathcal{H}^{2n+1}\cap \mathrm{GL}_{n+2}(\mathbb{Z})$.

The manifolds $M^{2k+1}_H$ are related by a sequence of embeddings of codimension 2:

$$ \begin{equation*} S^1\to M^3_H\to\cdots\to M^{2k-1}_H\to M^{2k+1}_H\to \cdots\,. \end{equation*} \notag $$
The manifolds $M^3$ and $M^3_H$ are diffeomorphic, and the diffeomorphism is induced by an algebraic isomorphism of the corresponding nilpotent Lie groups. For $k>1$ the manifolds $M^{2k+1}$ and $M^{2k+1}_H$ are different.

The manifold $M^{2n}$ has a canonical symplectic structure, and the manifolds $M^{2n+1}$ and $M^{2n+1}_H$ are contact manifolds. The manifold $M_H^{2n+1}$ is also the total space of the fibre bundle $M^{2n+1}_H \to \mathcal{V}^n$ associated with the canonical Kähler form of the $n$-dimensional Abelian variety $\mathcal{V}^n$.

In § 3 we develop the theory of the trigraded Buchstaber spectral sequence $\operatorname{Bss}$. The construction of $\operatorname{Bss}$ for an arbitrary graded algebra is presented in § 3.1. In § 3.2 the spectral sequence $\operatorname{Bss}$ is represented as a new functorial structure on the cohomology $\operatorname{Ext}_A^{*,*}(M,N)$ of a graded Hopf algebra $A$ over a ring $R$ with arbitrary $A$-modules $M$ and $N$. In § 3.3 it is shown that for $M=N=R$ the new structure defines an increasing filtration on $\operatorname{Ext}^{s,t}_A(R,R)$.

In § 3.4 we obtain an explicit formula for the differential $d_1$ by using the fact that for Hopf algebras over a field $\Bbbk$ the first term of $\operatorname{Bss}$ has the form

$$ \begin{equation*} E_1^{p,-(p+s),*}=(N_p/N_{p-1})\otimes\operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk). \end{equation*} \notag $$

In the case $R=\Bbbk$ we construct in § 3.5 an injective homomorphism from the algebra $\bigoplus\limits_p N_{p}/N_{p-1}$ to the tensor algebra $T(s\operatorname{Ext}^{1,*}_{A}(\Bbbk,\Bbbk))$ of the suspended vector space $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$ with shuffle product (see Theorem 14).

Based on this result, we introduce partial multivalued $\operatorname{Bss}$-operations on the tensor algebra $T(s \operatorname{Ext}^{1,*}_{A}(\Bbbk,\Bbbk))$ with values in $\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$. In § 3.6 it is shown that every element of the cohomology algebra $\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$ belongs to the image of a non-trivial $\operatorname{Bss}$-operation.

In § 3.7 we consider $\operatorname{Bss}$ in the case of the universal enveloping algebras of finite-dimensional nilpotent Lie algebras. Note that the Bss of the universal enveloping algebra for an ungraded Lie algebra is well defined and has all required properties, but it is bigraded, rather than trigraded.

In § 4 we review the example of the polynomial algebra in finitely many generators, considered as a Hopf algebra with comultiplication given by the condition that the generators are primitive. It is shown that in this example the new structure is completely described by the classical Koszul resolution (see Theorem 6).

In § 5 we compute fully the Buchstaber filtration in the Hopf algebra $UL_1$ and in the $S$-module $\Omega_*(\mathbb{Z})\otimes \mathbb{Q}=S_*\otimes \mathbb{Q}$ (see Theorems 7 and 10). In combination with Goncharova’s theorem, this yields a complete computation of the $E_1^{*,*,*}$-term of the corresponding $\operatorname{Bss}$.

In § 6.1 we collect the necessary information about higher matric Massey products, introduced in [13]. We follow the sign convention of [14] there.

A small digression is needed to explain the connection of §§ 6.2 and 6.3 with the subject of our article.

Soon after the publication of [12] and [11], where the notion of the Massey product in the cohomology $H^*(A,d_A)$ of a differential graded algebra was introduced, it was noticed that these operations are related directly to the differentials of certain spectral sequences.

Apparently, the first publication on this topic was [42]. The bar-construction of an $A(n)$-algebra $A$ was defined there: $\widetilde{\mathcal{B}}(A)=\bigoplus\limits_{i=0}^n\bar{A}^{\otimes i}$, and a spectral sequence was constructed from the filtration $F^p\widetilde{\mathcal{B}}(A)= \bigoplus\limits_{i\leqslant p}\bar{A}^{\otimes i}$. The differentials of this spectral sequences were interpreted as certain operations (called Yessam-operations by Stasheff) on the cohomology of $A$ with respect to the differential $\partial=m_1$. The interested reader will find all the necessary definitions and notation in [42].

There is a large number of papers where certain operations on the cohomology of algebras are associated with differentials of appropriate spectral sequences, and we do not aim to give an exhaustive overview of the results on this topic.

It was shown in [17] that differentials of the Eilenberg–Moore spectral sequence (EMss) can be expressed in terms of the so-called matric Massey products. This result was stated in [43], six years prior to the publication of [17], with the intention of providing detailed proofs in the article “The structure and applications of the Eilenberg–Moore spectral sequences”, which, however, was never published. The proof appeared in [17], § 5. The notion of the canonical Massey product is also introduced there, and it is proved that any cohomology class of dimension greater than $1$ of a connected graded algebra over a field can be expressed as an iterated matric Massey product of one-dimensional cohomology classes.

We also draw attention to the example of a connected and simply connected space $X$ constructed in [44]. Its cohomology contains non-trivial higher Massey products that are not determined by the differentials of $\operatorname{EMss}$ for the corresponding fibration $\Omega X\to PX\to X$. In § 6.2 we recall the algebraic construction of $\operatorname{EMss}$, and in § 6.3 we describe the known solutions of problems (1) and (2) using $\operatorname{EMss}$.

Higher structures on the cohomology of algebras, such as $A_\infty$-structures, have found numerous applications. In conclusion, we note that problems (1), (2), and (3) are important not only for Massey products but also for other types of higher structures on the cohomology of algebras.

In §§ 7 and 8 we consider Hopf algebras whose cohomology rings are isomorphic to the cohomology rings of the nilmanifolds $M^n$ and $M_H^{2n+1}$. These Hopf algebras are relatively simple, but nevertheless the calculation of the corresponding rings turns out to be non-trivial. For example, even the Betti numbers of nilmanifolds $M^n$ have not yet been calculated completely.

In § 7 we compute the $\operatorname{Bss}$ for the cohomology of the Lie algebras $\mathcal{G}L^3=\mathcal{GH}^3$ and $\mathcal{G}L^4$. We show that in the $\operatorname{Bss}$ of $\mathcal{G}L^n$ the differential $d_n$ is non-trivial, and all differentials with greater indices are trivial. We also show that for even $n$ the symplectic 2-form $\Omega_{n}$ on $M^n$ belongs to the image of the differential $d_n$.

In § 8 we give a survey of the results on the cohomology of the Heisenberg–Lie algebras $\mathcal{GH}^{2n+1}$, that is, the Lie algebras of the generalised Heisenberg groups ${\mathcal{H}}^{2n+1}$. Their Betti numbers were calculated in [18]. Following [45] and [46], we describe additive generators of $H^*(M_H^{2n+1})$ in terms of the $\mathfrak{sl}_2$-representation on the cohomology of the torus $T^{2n}$ equipped with a symplectic structure. A number of structure results on $\operatorname{Bss}$ for the Heisenberg–Lie algebras $\mathcal{GH}^{2n+1}$ are presented there, including the calculation of the $\operatorname{Bss}$ for $\mathcal{GH}^5$ and $\mathcal{GH}^7$.

It is shown that for the $\operatorname{Bss}$ of all examples from §§ 7 and 8 any element of the cohomology group lying in the image of the differential $d_k$ is represented by a $(k+1)$-fold non-trivial matric Massey product.

In § 9 the following problems are discussed:

(1) realisation of cohomology classes of Hopf algebras by Massey products;

(2) realisation of differentials in $\operatorname{Bss}$ by Massey products;

(3) effective construction of some special Massey products as differentials in $\operatorname{Bss}$.

The solution of problem (1) is based on the solution of problem (2). We prove that an element in the cohomology of a Hopf algebra of strict filtration $k$, that is, an element of the image of the differential $d_k$ modulo the differentials of lower order, can be represented as a non-trivial $(k+1)$-fold matric Massey product of special type (see Theorem 21). Given a cohomology class and its representing cocycle in the bar construction, this construction allows us to present the matric Massey product together with its defining system. Before we prove the general case, the mechanism for constructing a special Massey product realising the differential $d_k$ is illustrated in detail for $k=1,2,3$.

Recall that the $\operatorname{Bss}$ converges to the trivial module, and this guarantees that our filtration exhausts the cohomology of the Hopf algebra. As a simple consequence of Theorem 21, we obtain a new proof of the well-known result that any cohomology class of a graded Hopf algebra of dimension higher than 1 is represented by a non-trivial iterated matric Massey product of one-dimensional classes. Unlike the approach based on $\operatorname{EMss}$ (see § 6.3), this does not require any additional arguments.

The same approach allows us to prove a similar statement for some ungraded Hopf algebras, including the universal enveloping algebras of finite-dimensional nilpotent Lie algebras. In the latter case the $\operatorname{Bss}$ is bigraded and converges to zero, since the Buchstaber filtration of the dual of the universal enveloping algebra is exhaustive.

2. Applications of the Buchstaber spectral sequence ($\operatorname{Bss}$)

2.1. Complex cobordism

The trigraded spectral sequence under consideration was constructed in [25] and [32] for the calculation of the second term of the Adams–Novikov spectral sequence $\operatorname{Ext}^{s,t}_S(\mathbb{Z},\Omega_U^*)$. There are inclusions of rings $\Omega^*_U\subset\Omega_U^*(\mathbb{Z})\subset\Omega_U^*\otimes\mathbb{Q}$ compatible with the action of the Landweber–Novikov algebra. In addition, there is an isomorphism $S_*=\Omega_U^*(\mathbb{Z})$, also compatible with the action of $S$. A filtration in $S_*=\Omega_U^*(\mathbb{Z})$ starting with $N_0= \Omega_U^*$ was constructed in [25], together with an isomorphism $\operatorname{Ext}^{1,*}_S(\mathbb{Z},\Omega_U^*)=N_1/N_0$. We therefore obtain an exact sequence of Abelian groups

$$ \begin{equation*} 0\to \Omega_{2n}^U\to(N_1)_{2n}\to \operatorname{Ext}^{1,2n}_S(\mathbb{Z},\Omega_U^*)\to 0, \end{equation*} \notag $$
where $(N_1)_{2n}$ denotes the component of degree $2n$. Let $\Omega^{U,\rm fr}_*$ denote the bordism groups of quasicomplex manifolds with stably framed boundaries, introduced by Conner and Floyd [47]. The action of the algebra $A_U$ of stable operations in complex cobordism implies the homomorphism $\lambda\colon\Omega^{U,\rm fr}_*\to N_1$. Recall that the Adams invariant $e_2$ is well defined for elements of stable homotopy groups of spheres. In [47] a construction of the Adams invariant in terms of the group $\Omega^{U,\rm fr}_*$ was presented. The quotients $(N_1)_{4n}/\operatorname{im} \lambda$ were calculated in [25].

These ideas were further developed by N. V. Panov [33], who constructed a basis of the group $(N_1)_{2n}/\Omega^U_{2n}$ for all $n$ and calculated the elements of $\Omega_U^*(\mathbb{Z})$ in $\operatorname{im}\lambda$. As a consequence, a solution to the well-known problem of describing the relations between the Chern numbers of $(U,\rm fr)$-manifolds was obtained in [33].

The groups $\operatorname{Ext}^{1,*}_S(\mathbb{Z},\Omega_U^*)= \operatorname{Ext}_{A^U}^{1,*}(\Omega_U^*,\Omega_U^*)$ were calculated by Novikov in [29]. We briefly recall his approach following the notation of [29]. Novikov used the Riemann–Roch transformation $\lambda\colon U^*(-)\to k^*(-)$ and the augmentation $\nu\colon U^*(-)\to H^*(-,\mathbb{Z}/p)$. Here $k^*(-)$ denotes the connective complex $K$-theory represented by the spectrum $(k_n)$, where $\Omega^{2n}k_{2n}=BU\times\mathbb{Z}$. Then Novikov showed that the transformations $\lambda$ and $\nu$ are compatible with the action of the corresponding algebras of stable cohomology operations, namely, the algebra $A^U=U^*(MU)$ for complex cobordism, the Steenrod algebra $A_p$ for $\mathbb{Z}/p$-cohomology, and the algebra $A_\Psi^k$ generated by the Adams–Novikov operations in $K$-theory. As a result, Novikov constructed homomorphisms

$$ \begin{equation*} \begin{aligned} \, \widetilde\lambda\colon \operatorname{Ext}_{A^U}^{1,t}(\Omega_U^*,\Omega_U^*) &\to\operatorname{Ext}_{A_\Psi^k}^{1,t}(k^*(\rm pt),k^*(\rm pt)), \\ \widetilde\nu\colon \operatorname{Ext}_{A^U}^{1,t}(\Omega_U^*,\Omega_U^*)&\to \operatorname{Ext}_{A_p}^{1,t}(\mathbb{Z}/p,\mathbb{Z}/p). \end{aligned} \end{equation*} \notag $$
It was shown that $\widetilde\lambda$ is injective ([29], Corollary 9.1). By combining various methods, Novikov proved that for all $t\ne 4$ the homomorphism $\widetilde\lambda$ is an isomorphism, and for $t=4$ the cokernel of $\widetilde\lambda$ has order 2 ([29], §§ 9–11). This led to the explicit calculation of $\operatorname{Ext}_{A^U}^{1,t}(\Omega_U^*,\Omega_U^*)$.

The next filtration term, $N_2$, was calculated in [36]. The answer was presented in two forms. First, for each prime $p$, elements of $\Omega_U^*(\mathbb{Z})$ representing classes of $N_2\otimes\mathbb{Z}_{(p)}$ were described. These elements follow a general pattern for $p\geqslant 5$; the case $p=2$ and $p=3$ are exceptional. Second, an analogue of the well-known Stong–Hattori theorem was obtained for the group $N_2$, that is, the elements of $N_2$ were characterised by the condition of integrality of certain characteristic numbers in $K$-theory.

Using the methods of [25] one obtains the exact sequence

$$ \begin{equation} 0\to N_2/N_1 \to \operatorname{Ext}^{1,*}_S(\mathbb{Z},N_1/\Omega_U^*) \to \operatorname{Ext}^{2,*}_S(\mathbb{Z},\Omega_U^*). \end{equation} \tag{2} $$
The proof essentially consists in analysing the $(-2)$th row of $\operatorname{Bss}$ for $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\Omega_U^*)$. The exact sequence (2) was presented in [36]. With its help, the non-triviality of a number of elements in the stable homotopy groups of spheres was established by using Massey products and elements with non-trivial Arf-invariant and not-rivial differential $d_2$ of the Buchstaber spectral sequence.

The results of [25], [32], [33], [36], and [35] were the first steps towards solving the problem of the geometric realisation of elements of the groups $\operatorname{Ext}^{*,*}_S(\mathbb{Z},\Omega_U^*)$. This problem is closely related to the well-known problem of the interpretation of the Adams–Novikov filtration for the sphere spectrum in terms of the Thom spaces classifying cobordism classes of manifolds with corners. For results in this direction, see [48].

2.2. Cohomology of the Lie algebra $L_1$ of formal vector fields on the line

The following example links the Landweber–Novikov algebra to the theory of infinite-dimensional Lie algebras (see [34]). Consider the Lie algebra $L_1$ of formal vector fields on the line that vanish at the origin together with their first derivative. This Lie algebra has a basis of vector fields $e_i= t^{i+1}\,\dfrac{d}{dt}$ , $i\geqslant 1$. It is easy to see that $[e_i,e_j]=(j-i) e_{i+j}$. The universal enveloping algebra $UL_1$ over the field $\mathbb{Q}$ is the associative algebra with unity generated by the symbols $e_i$, $i \geqslant 1$, with the relations $e_ie_j -e_j e_i=(j-i) e_{i+j}$ for $i\ne j$.

According to Goncharova’s theorem [37], [38] all positive (first) graded components of the cohomology of $L_1$ are two-dimensional. Furthermore, if the first grading is equal to $s$, then the corresponding generators have a second grading equal to the so-called Euler pentagonal numbers $(3s^2\pm s)/2$. The proof of this theorem has been improved consistently (see [49]–[54]).

Let $x^s_\pm$ denote the two generators of the group $\operatorname{Ext}_{UL_1}^{s,(3s^2\pm s)/2}(\mathbb{Q},\mathbb{Q})$. The sum of two pentagonal numbers is never a pentagonal number, which implies that multiplication in the cohomology of $L_1$ is trivial. In the early 1970s Buchstaber conjectured that the whole cohomology algebra $\operatorname{Ext}_{UL_1}^{*,*}(\mathbb{Q},\mathbb{Q})$ is generated by non-trivial (that is, not containing zero) iterated Massey products of the two generators $x^1_\pm\in \operatorname{Ext}_{UL_1}^{1,*}(\mathbb{Q},\mathbb{Q})$. A number of results have been obtained in this direction. The technique developed in [55] made it possible to prove that

$$ \begin{equation*} x^s_-\in\langle x^{s-1}_+,\underbrace{ x^1_-,\dots,x^1_-}_{2s-1}\,\rangle \quad\text{and}\quad x^s_+\in\langle x^{s-1}_+,\underbrace{ x^1_-,\dots,x^1_-}_{3s-1}\,\rangle. \end{equation*} \notag $$
However, it was soon noticed that all these products contain zero. In the note [56] by Artelnykh it was stated that the elements $x^s_-$ and $x^{2s+1}_+$ are realised by non-trivial Massey products:
$$ \begin{equation*} x^s_-\in\langle \underbrace{ x^1_+,\dots,x^1_+}_{s-1},x^{s-2}_+, x^1_-\rangle \quad\text{and}\quad x^{2s+1}_+\in\langle \underbrace{ x^1_+,\dots,x^1_+}_{3s+1}, x_+^{2s}\rangle. \end{equation*} \notag $$
The method of [56] uses the connection between special modules and non-trivial higher differentials in $\operatorname{Bss}$. Unfortunately, detailed proofs of the results in [56] have not yet been published. Buchstaber’s conjecture was finally proved by Millionshchikov [57] in 2009. He showed that
$$ \begin{equation} \begin{aligned} \, x^{s+1}_-&\in \langle\underbrace{ x^1_-,\dots,x^1_-}_{m}, x^1_+, \underbrace{ x^1_-,\dots,x^1_-}_{n}, x^{s}_+\rangle, \\ x^{s+1}_+&\in \langle \underbrace{ x^1_-,\dots,x^1_-}_{s}, x^1_+, \underbrace{ x^1_-,\dots,x^1_-}_{2s}, x^{s}_+ \rangle, \end{aligned} \end{equation} \tag{3} $$
where $n$, $m$ are arbitrary natural numbers, $m+n=2s-1$, and all these Massey products are non-trivial. Note that all Massey products representing elements $x_{\pm}^{s+1}$ are special, but Millionshchikov’s result is stronger than the general result of Theorem 21 and Corollary 11 in our paper, since in [57] he used only scalar Massey products, rather than matric ones. Therefore, a natural problem arises: describe the classes of Hopf algebras and Lie algebras for which Theorem 21 and Corollary 11 can be strengthened by using only scalar Massey products.

2.3. Nilmanifolds of polynomial transformations of the line

Massey products are related to the tower of nilmanifolds introduced in [39], which we discuss in detail in § 7. The results presented below in this section were obtained in [34]. Consider the group $\operatorname{Diff}^1(\mathbb{R})$ of formal diffeomorphisms of the line that leave zero fixed and have a derivative equal to one at zero. The subgroup $\operatorname{Diff}^1(\mathbb{Z})$ consists of all diffeomorphisms with integer coefficients. It turns out that the Landweber–Novikov algebra $S$ is isomorphic to the algebra of left-invariant differential operators on the group $\operatorname{Diff}^1(\mathbb{Z})$. Furthermore, the Landweber–Novikov algebra tensored with reals is isomorphic to the universal enveloping algebra of the Lie algebra of formal vector fields on the line that vanish at zero together with the first derivative.

All these facts are proved within the theory of generalised shift operators (see [58]). Let $G$ be a group, and let $\mathcal F$ be a subring of functions on $G$ which is closed under the shift operator

$$ \begin{equation*} T_x^y\colon f(x)\mapsto f(xy)=F(x,y). \end{equation*} \notag $$
Then we obtain a ring homomorphism
$$ \begin{equation*} T_x^y\colon \mathcal F\to\mathcal F\mathbin{\widehat\otimes}\mathcal F,\quad T_x^y (f(x))=F(x,y),\quad\text{where}\ \ f(x)\in \mathcal F\ \ \text{and}\ \ F(x,y)\in\mathcal F\mathbin{\mathbin{\widehat\otimes}} \mathcal F, \end{equation*} \notag $$
satisfying the associativity equation $T_y^zT_x^y f(x)=T_x^yT_x^z f(x)$.

In the case of the additive group $G=\mathbb{R}^n$ we obtain the shift $T^y_x f(x)=f(x+y)$, which defines the Taylor series in $y$. In the case of an arbitrary group we obtain an analogue of a Taylor series, which is used to construct left-invariant operators $L_x$ on $\mathcal F$. These operators satisfy the identity

$$ \begin{equation*} L_x( T_x^y (f(x)))=T_x^y(L_x f(x)). \end{equation*} \notag $$
It can be proved that they generate the ring of left-invariant operators on $\mathcal F$.

Note that in the theory of generalised shift operators (GSO) the operator $T^y_x$ is assumed to be linear, but it need not be a ring homomorphism. Other properties of this operator are taken as axioms of the theory of generalised shift operators. For example, in [59] it was shown that every $n$-valued group defines a GSO on its ring of functions.

The set $\operatorname{Diff}^1(\mathbb{Z})$ is a subgroup with respect to multiplication in the group $\operatorname{Diff}^1(\mathbb{R})$. The rings of function on $\operatorname{Diff}^1(\mathbb{R})$ and $\operatorname{Diff}^1(\mathbb{Z})$ are the polynomial rings $P(\mathbb{R})$ and $P(\mathbb{Z})$ in infinitely many variables over $\mathbb{R}$ and $\mathbb{Z}$, respectively. In this case the linear operators $L_x$ are differential operators on these rings. The fact that the Landweber–Novikov algebra $S$ is isomorphic to the algebra of left-invariant differential operators on the ring $P(\mathbb{Z})$ is verified directly using the ring isomorphism $P(\mathbb{Z})\cong\Omega^*_U(\mathbb{Z})$ and the remarkable fact that the set of all multiplicative operations in complex cobordism is isomorphic to the group $\operatorname{Diff}^1(\mathbb{Z})$.

If we consider only polynomial transformations of degree at most $n+1$ in $\operatorname{Diff}^1$ and reduce their composition modulo $t^{n+2}$, then we obtain an interesting nilpotent Lie group $L^n$ (see [39], [40], and [14]). Its Lie algebra $\mathcal{G}L^n$ is isomorphic to the quotient of the algebra $L_1$ by the ideal generated by the elements $e_k=t^{k+1}\,\dfrac{d}{dt}$ , $k\geqslant n+1$. Polynomial transformations with integer coefficients form a lattice in $L^n$. The quotient of $L_n$ by this lattice is a nilmanifold, which we denote by $M^n$. The manifolds $M^n$, $n\geqslant1$, form a tower of bundles

$$ \begin{equation} \cdots \to M^n\to M^{n-1}\to \cdots \to M^2\to M^1=S^1 \end{equation} \tag{4} $$
with fibre $S^1$. The nilmanifolds $M^n$ have remarkable properties, which have found numerous applications. In particular, the real cohomology ring of $M^n$ has an additional second grading, which is natural with respect to the tower projections in (4). The second grading allowed the first-named author to define in [60] the polynomial Euler characteristic of manifolds $M^n$ and prove its multiplicativity with respect to the tower of bundles (4). Note that the ordinary Euler characteristic of each manifold $M^n$ is zero.

In [61] and [62] the well-known problem of constructing explicit examples of rationally nonformal simply connected symplectic manifolds was solved. These results were further developed in the work [14] by the same authors, where it was shown that simply connected symplectic manifolds with non-trivial $2n$-fold Massey products exist for arbitrarily large $n$. The manifold $M^{2n}$ admits a symplectic structure, given by the 2-form

$$ \begin{equation*} \Omega_{2n}=\sum_{i<n}(2n-2i)\omega_{2n-i}\wedge \omega_i \end{equation*} \notag $$
(see [39], [40], and [14]). It was proved in [14] that the cohomology class $[\Omega_{2n}]\in H^2(M^{2n})$ represents a non-trivial $2n$-fold Massey product and does not represent an $m$-fold Massey product with $m<2n$. The proof of the last statement uses bigrading and an auxiliary filtration.

2.4. Heisenberg nilmanifolds

Another important series of nilmanifolds arises from the generalisation of the Heisenberg group ${\mathcal H}^3$ to higher dimensions. Consider the group ${\mathcal H}^{2n+1}$ of matrices of the form

$$ \begin{equation*} \begin{pmatrix} 1 & x_1 & x_2 & \dots & x_n & z \\ 0 & 1 & 0 & \dots & 0 & y_1 \\ &\ddots&\ddots&\ddots&\vdots &\vdots \\ && \ddots&1 & 0& y_{n-1} \\ &{\large\textbf{0}} & & 0 & 1& y_n \\ &&& & 0 & 1 \end{pmatrix} \end{equation*} \notag $$
with real $x_j$, $y_j$, and $z$. As a space, it is homeomorphic to $\mathbb{R}^{2n+1}$. The group ${\mathcal H}^{2n+1}$ contains the lattice $\Gamma_H^{2n+1}$ consisting of matrices with integer $x_j$, $y_j$, and $z$. The corresponding Lie algebra $\mathcal{GH}^{2n+1}$ is nilpotent; hence the quotient $M_H^{2n+1}={\mathcal H}^{2n+1}/\Gamma_H^{2n+1}$ is a nilmanifold.

In dimension 3 there are only two nilpotent Lie algebras, one of which is Abelian, so the Lie groups $L^3$ and $\mathcal{H}^3$ are clearly isomorphic. We present an isomorphism $F\colon L^3\to \mathcal{H}^3$ of these groups that restricts to an isomorphism of the lattices chosen, and thus defines a diffeomorphism $M^3\cong M_H^3$. As a space $\mathcal{H}^3$ is diffeomorphic to $\mathbb{R}^3$. To a point $(u_1,u_2,u_3)\in \mathbb{R}^3$ we assign the matrix $\begin{pmatrix}1& u_1 & u_3\\ 0&1&u_2\\0&0&1\end{pmatrix}$. The product of two matrices $\begin{pmatrix}1& u_1 & u_3\\ 0&1&u_2\\0&0&1\end{pmatrix}$ and $\begin{pmatrix}1& v_1 & v_3\\ 0&1&v_2\\0&0&1\end{pmatrix}$ from $\mathcal{H}^3$ is a matrix of the same form $\begin{pmatrix}1& w_1 & w_3\\ 0&1&w_2\\0&0&1\end{pmatrix}$, where

$$ \begin{equation} w_1= u_1+v_1,\qquad w_2=u_2+v_2,\quad\text{and}\quad w_3=u_3+v_3+u_1v_2. \end{equation} \tag{5} $$

The group $L^3$ is $\mathbb{R}^3$ with group structure defined as follows (see [40] and § 7). We put

$$ \begin{equation*} L^3=\{p_x(t)=t+x_1t^2+x_2 t^3+x_3 t^4\colon x_1,x_2,x_3\in\mathbb{R}\}. \end{equation*} \notag $$
The product of two elements $L^3$ is defined as a composition:
$$ \begin{equation*} (p_x * p_y)(t)=p_z(t)=p_y(p_x(t))\mod t^{5}, \end{equation*} \notag $$
which is explicitly given by
$$ \begin{equation} z_1=x_1+y_1, \qquad z_2 =2 x_1 y_1 +x_2 +y_2, \qquad z_3 =x_3+ (x_1^2+2 x_2)\, y_1+3 x_1 y_2+y_3. \end{equation} \tag{6} $$
These formulae are clearly different from (5).

Consider the map $F\colon L^3=\mathbb{R}^3\to \mathbb{R}^3=\mathcal{H}^3$, defined by the formulae

$$ \begin{equation} u_1=x_1, \qquad u_2=x_2-x_1^2,\quad\text{and}\quad u_3=x_3-2x_2x_1+x_1^3. \end{equation} \tag{7} $$
It is easy to see that $F$ is invertible, and the inverse map is also given by polynomials. A direct check shows that the transformation $v=F(y)$, $w=F(z)$ takes formulae (5) to (6). That is, $F$ is a homomorphism (see [60], Lemma 3.1).

Finally, we note that for $n\geqslant 2$ the groups $L^{2n+1}$ and $\mathcal{H}^{2n+1}$ are not isomorphic, since their Lie algebras have different degree of nilpotency.

3. Construction and properties of $\operatorname{Bss}$

3.1. Graded algebras

Let $A$ be a graded algebra over a base ring $R$ with unity, and assume that $A=\bigoplus\limits_{n\geqslant 0} A^n$, where each homogeneous component is a finitely generated $R$-module and $A^0=R$. It is convenient to assume that $R$ is graded trivially, that is, concentrated in degree zero.

Recall the definition of $\operatorname{Ext}^{s,t}_A(M,N)$, where $M$ and $N$ are left graded $A$-modules, that is, $A^i M^j \subset M^{i+j}$. Let $\cdots \to P_2\to P_1 \to P_0\to M $ be a projective resolution of the $A$-module $M$. The group $\operatorname{Hom}_A^t(M,N)$ consists of the $A$-linear homomorphisms that raise the grading by $t$. Applying the functor $\operatorname{Hom}_A^t(-,N)$ to the resolution, we obtain the complex $\operatorname{Hom}^t_A(P_s,N)$ with differential $d$ induced by the differential of the resolution, which preserves the $t$-grading and raises the $s$-grading by $1$. The cohomology of this complex with respect to $d$ is denoted by $\operatorname{Ext}^{s,t}_A(M,N)$.

Definition 1. Let $N$ be an $A$-module, and let $N_0$ be its $A$-invariant $R$-submodule. The increasing Buchstaber filtration on $N$ is defined recursively:

$$ \begin{equation} N_{p+1} =\biggl\{n\in N\colon s\cdot n\in N_p \text{ for all } s\in A^+=\bigoplus_{n>0} A^n\biggr\}. \end{equation} \tag{8} $$

The main property of this filtration is that the action of elements of $A^+$ on the quotients $N_{p+1}/N_{p}$ is trivial for $p\geqslant 1$, and the unity of the ring $A$ acts identically on $N_{p+1}/N_{p}$.

The Buchstaber filtration induces an increasing filtration of the complex $(\operatorname{Hom}^t_A(P_s,N),d)$ by the subcomplexes $\operatorname{Hom}^t_A(P_s,N_p)$. Since it is compatible with the differential, a trigraded spectral sequence $\operatorname{Bss}$ arises. It has the following properties:

1. $E_1^{p,q,t}=\operatorname{Ext}_A^{-(p+q),t}(M,N_p/N_{p-1})$. Moreover, the groups $\operatorname{Ext}_A^{-(p+q),t}(M,N_p/N_{p-1})$, $\operatorname{Ext}_A^{-(p+q),t}(M,R)$ and $N_p/N_{p-1}$ are related by the universal coefficient theorem. For $p\geqslant 1$ this relation is simplified, since then the action of $A$ on $N_p/N_{p-1}$ is trivial. In particular, if $R$ is a field, then

$$ \begin{equation*} E_1^{p,q,t} =\bigoplus_{t=t_1+t_2}\operatorname{Ext}_A^{-(p+q),t_1}(M,R) \otimes(N_p/N_{p-1})^{t_2}. \end{equation*} \notag $$

2. $E_r^{*,*,*}$ for $r\geqslant 1$ does not depend on the choice of projective resolution

$$ \begin{equation*} \cdots \to P_2\to P_1 \to P_0\to M. \end{equation*} \notag $$

3. The differential $d_1$ is the connecting homomorphism in the long exact sequence of the triple of complexes corresponding to the inclusions $N_{p-2}\subset N_{p-1}\subset N_p$:

$$ \begin{equation*} \operatorname{Ext}_A^{-(p+q),t}(M,N_p/N_{p-1})\to \operatorname{Ext}_A^{-(p+q)+1,t}(M,N_{p-1}/N_{p-2}). \end{equation*} \notag $$
In particular, $d_1$ reduces the $p$-grading by one and preserves the $q$- and $t$-gradings:
$$ \begin{equation*} d_1\colon E_1^{p,q,t} \to E_1^{p-1,q,t}. \end{equation*} \notag $$

4. The differential $d_r$ changes the gradings as follows:

$$ \begin{equation*} d_r\colon E_r^{p,q,t}\to E_r^{p-r,q+r-1,t}. \end{equation*} \notag $$

5. Taking into account the restrictions on the grading of $A$ and the filtration on $N$, we obtain $E_1^{p,q,t}=0$ for $p<0$ and for $p+q>0$.

6. $E^{0,q,t}_1=\operatorname{Ext}^{-q,t}_A(M,N_0)$ (by the construction of the filtration).

7. $E^{p,-p,t}_1= \operatorname{Hom}^{t}_A (M,N_p/N_{p-1})$ (by the construction of the filtration and the definition of $\operatorname{Ext}^{*,*}_A$).

8. If the filtration $N_p$ satisfies the standard conditions (for example, if each grading is exhausted in a finite number of filtration steps), then the spectral sequence converges to $\operatorname{Ext}^{s,t}_A(M,N)$.

We introduce the following terminology. A submodule $N_0\subset N$ is called

$\bullet$ trivial if $N_1=N_0$, which implies that $N_p=N_0$ for $p>1$;

$\bullet$ complete if the filtration $N_p$ exhausts $N$.

Applications of the Buchstaber spectral sequence are based on the following considerations. Suppose we need to compute $\operatorname{Ext}^{s,t}_A(M,N_0)$. We find an extension $N$ of the module $N_0$ such that $\operatorname{Ext}^{s,t}_A(M,N)$ is known, for example, trivial. If $N_0\subset N$ is a complete submodule, then $\operatorname{Bss}$ converges to $\operatorname{Ext}^{s,t}_A(M,N)$, and its initial term contains $\operatorname{Ext}^{s,t}_A(M,N_0)$ as the zero column.

On the contrary, if the module $N_0\subset N$ is trivial, then the spectral sequence degenerates and cannot be used to obtain information about $\operatorname{Ext}^{s,t}_A(M,N_0)$ from $\operatorname{Ext}^{s,t}_A(M,N)$.

In the case of graded Hopf algebras which is under consideration, the module $N_0=R\subset A_*=N$ is always complete, which can easily be seen from the grading. In the context of Lie algebras (see § 3.7) the Buchstaber filtration is defined on the dual of the universal enveloping algebra of a Lie algebra. In this case, the $\operatorname{Bss}$ can also be constructed, but it is bigraded rather than trigraded, since the universal enveloping algebra is an ungraded Hopf algebra. The completeness of the module $N_0=R$ is equivalent to the residual nilpotency of the Lie algebra. The module $N_0= \mathbb{R}$ is trivial, for example, in the case of the universal enveloping algebra of $\mathfrak{so}(3)$.

3.2. Graded Hopf algebras

Now suppose that $A$ is a bialgebra, and let $A_*$ be its dual. In all interesting examples $A$ is, in fact, a Hopf algebra, so we will regularly speak of Hopf algebras, although the constructions in this section do not use the antipodal map (recall that a Hopf algebra is a bialgebra equipped with an antipodal map; see, for example, [8]). Let $N_0\subset A_*$ be an invariant submodule of $A$ containing $1\in A_*$. In applications we often have $N_0=R\subset A_*$. It is easy to verify that the restrictions on the grading imply that $N_0$ is a complete submodule. Consider the Buchstaber filtration $N_0\subset N_1\subset\cdots\subset A_*$. From the definition of the action of $A$ on $A_*$ it follows that the inductive definition (8) of the submodules $N_p$ can be reformulated as follows:

$$ \begin{equation} N_{p+1}=\Bigl\{x\in A_*\colon x^{(1)}_i\in N_p \text{ for all }i, \text{ where }\Delta x=x\otimes 1+\sum x^{(1)}_i\otimes x^{(2)}_i\Bigr\}. \end{equation} \tag{9} $$

The (reduced) two-sided bar construction [63] is the complex of $A$-bimodules $(B_{\small\bullet}(A,A,A),d_B)$, where $B_{s}(A,A,A)=A\otimes (\bar{A})^{\otimes s}\otimes A$ and $\bar{A} =A/R$ is the augmentation ideal. The differential $d_B$ is given by

$$ \begin{equation*} d_B(a_0\otimes a_1\otimes \cdots\otimes a_{n+1})= \sum_{k=0}^n (-1)^{\varepsilon_k} a_0\otimes \cdots \otimes a_ka_{k+1} \otimes\cdots\otimes a_{n+1}, \end{equation*} \notag $$
where $(-1)^{\varepsilon_k}$ is some sign, which we will specify later.

The complex $(B_{\small\bullet}(A,A,A),d_B)$ is a free resolution of the $A$-bimodule $A$. Furthermore, $B_{\small\bullet}(A,A,M)= B_{\small\bullet}(A,A,A)\otimes_A M $ is a free $A$-resolution of an $A$-module $M$. In particular, $B_{\small\bullet}(A,A,R)=B_{\small\bullet}(A,A,A)\otimes_\varepsilon R$ is a free resolution of the base ring $R$ with the $A$-module structure given by the augmentation $\varepsilon\colon A\to R$.

Let $A_*$ be the dual Hopf algebra with diagonal $\Delta$. We define the maps $\widehat\Delta$ and $\overline\Delta$ by $\widehat\Delta(x)=\Delta(x)-x\otimes 1$ and $\overline\Delta(x)=\Delta (x)-x\otimes 1-1\otimes x$.

Then $\operatorname{Hom}^*_{A}(B_{\bullet}(A,A,R),A_*)$ is a one-sided cobar construction for $A_*$, namely, the complex of modules $F^s(A_*,A_*,R)= A_*\otimes (\bar{A}_*)^{\otimes s}$ with differential

$$ \begin{equation} \begin{aligned} \, &d_F\colon F^s(A_*,A_*,R)\to F^{s+1}(A_*,A_*,R), \\ &d_F(x_0\otimes x_1\otimes \cdots \otimes x_s)=1\otimes x_0\otimes x_1\otimes \cdots \otimes x_s \\ &\qquad+(-1)^{| x^{(1)}_{0,j}|} x^{(1)}_{0,j}\otimes x^{(2)}_{0,j} \otimes x_1\otimes \cdots \otimes x_s \\ &\qquad+\sum_{k=1}^s \sum_j(-1)^{\varepsilon_k} x_0\otimes \cdots \otimes x^{(1)}_{k,j}\otimes x^{(2)}_{k,j}\otimes \cdots \otimes x_s, \end{aligned} \end{equation} \tag{10} $$
where
$$ \begin{equation*} \Delta x_k=x_k\otimes 1+1\otimes x_k + \sum_j x^{(1)}_{k,j}\otimes x^{(2)}_{k,j}\quad\text{and}\quad \varepsilon_k=|x_0|+\cdots+| x_{k-1}|+|x^{(1)}_{k,j}|-k. \end{equation*} \notag $$

The complex $F^{\small\bullet}(R,A_*,R)$ computes the groups $\operatorname{Ext}^{*,*}_A(R,R)$ and embeds in $F^{\small\bullet}(A_*,A_*,R)$ as a subcompex. Under this inclusion, an element $x_1\otimes\cdots\otimes x_s \in F^{s}(R,A_*,R)$ is mapped to $1\otimes x_1\otimes\cdots\otimes x_s \in F^{s}(A_*,A_*,R)$.

It is also easy to verify that the complex

$$ \begin{equation*} \operatorname{Hom}^*_{A}(B_{\small\bullet}(A,A,R),N_p) \subset \operatorname{Hom}^*_{A}(B_{\small\bullet}(A,A,R),A_*) \end{equation*} \notag $$
is isomorphic to the subcomplex
$$ \begin{equation*} F^{\small\bullet}(N_p,A_*,R)\subset F^{\small\bullet}(A_*,A_*,R). \end{equation*} \notag $$
Thus, the filtration and the corresponding trigraded Buchstaber spectral sequence can also be constructed using the cobar resolution.

Lemma 1. Suppose that the base ring $R$ is a field $\Bbbk$. Then the first sheet of the trigraded spectral sequence associated with the filtration $N_0\subset N_1\subset \cdots\subset A_*$ is isomorphic to the cohomology of the complex $\bigoplus\limits_p N_p/N_{p-1} \otimes F(\Bbbk, A_*,\Bbbk)$ with respect to the differential $d_F$ in $F(\Bbbk, A_*,\Bbbk)$.

Proof. In fact, the first sheet of the trigraded spectral sequence is nothing but the cohomology of the quotient complex $F(N_p, A_*,\Bbbk)/F(N_{p-1},A_*,\Bbbk)$. It remains to note that by the definition of the Buchstaber filtration the first term in formula (10) for $d_F$ vanishes on the quotient complex. $\Box$

Let $A$ be a bialgebra over a ring $R$, and let $N_0\subset A_*$ be an $A$-invariant submodule, $1\in N_0$. Consider the Buchstaber filtration $N_p$ in $A_*$ starting from $N_0$.

Lemma 2. Suppose that $N_0$ is a subring of $A_*$. Then the filtration $N_p$ is multiplicative, that is, if $x\in N_i$ and $y\in N_j$, then $xy\in N_{i+j}$.

Proof. We use induction on $p+q$. The induction base $N_0\cdot N_0\subset N_0$ is true by assumption. Suppose that $y\in N_p$ and $x\in N_q$. Let
$$ \begin{equation*} \Delta x=x\otimes 1 +\sum x^{(1)}_i\otimes x^{(2)}_i,\qquad \Delta y=y\otimes 1 +\sum y^{(1)}_j\otimes y^{(2)}_j. \end{equation*} \notag $$
Then
$$ \begin{equation*} \Delta (xy)=xy\otimes 1+\sum xy^{(1)}_j\otimes y^{(2)}_j + \sum \pm x^{(1)}_i y \otimes x^{(2)}_i+ \sum \pm x^{(1)}_i y^{(1)}_j\otimes x^{(2)}_i y^{(2)}_j. \end{equation*} \notag $$
It remains to note that for all terms except the first, the first factor in the tensor product has filtration degree at most $p+q-1$ by the induction hypothesis. $\Box$

Now suppose that $R$ is a field, denoted by $\Bbbk$. Let $I$ be the augmentation ideal of $A$. Consider the filtration of $A$ by the powers of $I$. Namely, let

$$ \begin{equation*} F^0=F^0A=A,\qquad F^1=F^1A=I,\qquad F^p=F^pA=I^p. \end{equation*} \notag $$
Clearly, $F^0/F^1=\Bbbk$. The module $F^1/F^2$ is called the module of indecomposable elements of $A$. The filtration by the powers of the augmentation ideal in a Hopf algebra was introduced in [64].

Lemma 3. Consider the Buchstaber filtration in $A_*$, which starts with $N_0=\Bbbk\subset A_*$.

The filtration $N_p(A_*)$ is dual to the filtration $F^p(A)$ in the following sense: the module $N_p(A_*)$ consists of those linear functions on $A$ that vanish on all elements of $I^{p+1}= F^{p+1}(A)$.

In particular,

$$ \begin{equation*} \begin{aligned} \, N_p(A_*)/N_{p-1}(A_*) &= (F^{p}(A)/F^{p+1}(A))^*, \\ \dim N_p(A_*)/N_{p-1}(A_*) &=\dim F^{p}(A)/F^{p+1}(A). \end{aligned} \end{equation*} \notag $$

Proof. We have $N_0(A_*)=\Bbbk$, and it consists of precisely the functions on $A$ that vanish on $I$. Next we use induction. By the induction assumption, $\phi\in N_{p+1}(A_*)$ if and only if for all $u\in I$ the function $u\cdot \phi$ belongs to $N_{p}(A_*)$. This means precisely that $x\mapsto \phi(xu)$ vanishes for any $x\in I^{p+1}$, in other words, $\phi(I^{p+2})=0$. $\Box$

3.3. Filtration in $\operatorname{Ext}_A^{*,*}(R,R)$

Let $A$ be a Hopf algebra over a ring $R$. The zero column $E_1^{0,-q,*}$ consists of the groups $\operatorname{Ext}_A^{q,*}(R,R)$, and for dimensional reasons all differentials on the groups $E_r^{0,-q,*}$ are trivial. Since $E_\infty^{0,-q,*}=0$, $q\geqslant 1$, the zero column of the trigraded spectral sequence must be exhausted by the images of the differentials $d_r$, $r\geqslant 1$, in the following sense: there is an increasing filtration $\Phi^r=\Phi^r \operatorname{Ext}_A^{q,*}(R,R)$ on $\operatorname{Ext}_A^{q,*}(R,R)=E_1^{0,-q,*}$ such that

(i) $\Phi^0=R=E^{0,0,0}_\infty$;

(ii) the quotient $\Phi^r/\Phi^{r-1} \subset \operatorname{Ext}_A^{q,*}(R,R)/\Phi^{r-1}$ coincides with the image of $d_r\colon E_r^{r,*-r+1,*}\to E_r^{0,*,*}$;

(iii) $\operatorname{Ext}_A^{q,*}(R,R)/\Phi^{r-1}=E_r^{0,-q,*}$.

Let $R$ be the field $\Bbbk$; then applying the formula for $d_1$ from Theorem 2 below to

$$ \begin{equation*} N_1/N_0\otimes \operatorname{Ext}^{q-1,*}_A(\Bbbk,\Bbbk) \xrightarrow{d_1} \operatorname{Ext}^{q,*}_A(\Bbbk,\Bbbk), \end{equation*} \notag $$
we obtain that $\Phi^2 \operatorname{Ext}_A^{q,*}(\Bbbk,\Bbbk)$ consists precisely of elements representable as $\displaystyle\sum a_i b_i$, where $a_i\in \operatorname{Ext}_A^{1,*}(\Bbbk,\Bbbk)$, $b_i\in \operatorname{Ext}_A^{q-1,*}(\Bbbk,\Bbbk)$.

The filtration $\Phi^r$ is natural in the following sense.

Theorem 1. Let $f\colon A\,{\to}\,A'$ be a homomorphism of bialgebras, and let $f^*\colon A'_*\,{\to}\,A_*$ be the dual homomorphism. Let $N_0\subset A_*$ be an $A$-invariant submodule, let $N'_0\subset A'_*$ be an $A'$-invariant submodule, and let $f^*(N'_0)\subset N_0$. Then the homomorphism $f^*$ is compatible with the Buchstaber filtrations on $A_*$ and $A'_*$ starting from $N_0$ and $N'_0$, respectively. That is, $f^*(N'_p)\subset N_p$.

Therefore, a homomorphism of the corresponding trigraded spectral sequences is defined. In particular,

$$ \begin{equation*} f^*(\Phi^r \operatorname{Ext}_{A'}^{q,*}(R,R))\subset \Phi^r \operatorname{Ext}_A^{q,*}(R,R). \end{equation*} \notag $$

Proof. The inclusion $f^*(N'_p)\subset N_p$ is easily proved by induction on $p$. Indeed, let $x\in N'_p$, so that all elements $x_1^{(i)}$ in the equality $\overline\Delta(x)=\displaystyle\sum_i x_i^{(1)}\otimes x_i^{(2)}$ belong to $N'_{p-1}$. Then
$$ \begin{equation*} \overline\Delta f^*(x)=f^*(\overline \Delta x)= \sum_i f^*(x_i^{(1)})\otimes f^*(x_i^{(2)}). \end{equation*} \notag $$
By the induction hypothesis all elements in the above sum satisfy $f^*(x_i^{(1)})\in N_{p-1}$ for all $i$, and therefore $ f^*(x) \in N_p$. $\Box$

3.4. The differential $d_1$

We briefly return to the case of Hopf algebras over an arbitrary ring $R$. Recall that the differential $d_1\colon E_1^{p,q,*}\to E_1^{p-1,q,*}$ is the connecting homomorphism in the long exact sequence of the triple of complexes

$$ \begin{equation*} F(N_{p-2}, A_*,R)\subset F(N_{p-1}, A_*,R)\subset F(N_p, A_*,R). \end{equation*} \notag $$
For $q=-1$ only $d_1\colon E_1^{1,-1,*}\to E_1^{0,-1,*}$ can be non-trivial, and it is easy to describe:

Lemma 4. The differential $d_1\colon E_1^{1,-1,*}\to E_1^{0,-1,*}$ coincides with the standard isomorphism $\delta\colon P(A_*)\to \operatorname{Ext}_A^{1,*}(R,R)$.

Proof. We have $E_1^{1,-1,*}=N_1/N_0$. By the definition of the filtration the module $N_1/N_0$ is naturally identified with the module of primitive elements
$$ \begin{equation*} P(A_*)=\{x\in A_*\colon \Delta x=1\otimes x + x\otimes 1\}. \end{equation*} \notag $$
Further, $E_1^{0,-1,*}=\operatorname{Ext}_A^{1,*}(R,R)$, and this module is isomorphic to $P(A_*)$. Considering representing cocycles, it is easy to check that $d_1$ maps $x\in N_1/N_0$ to $1\otimes x \in F(R, A_*,R)$, so $d_1$ is an isomorphism. $\Box$

Corollary 1 ([25], [36]). The differential $d_1\colon N_2/N_1=E_1^{2,-2,*}\to E_1^{1,-2,*}$ is injective.

Proof. By dimensional reasons, the only non-trivial differentials $d_r\colon E_{r}^{2,-2,*}\to E_{r}^{2-r,-3+r,*}$ are $d_1$ and $d_2$. Lemma 4 implies that $d_2\colon E_{2}^{2,-2,*} \to E_{2}^{0,-1,*}$ takes values in the zero group, so $E_{2}^{2,-2,*}=E_{\infty}^{2,-2,*}=0$, and so $d_1\colon N_2/N_1=E_1^{2,-2,*}\to E_1^{1,-2,*}$ is injective. $\Box$

In what follows we assume that $R$ is a field $\Bbbk$. The by Lemma 1 there is an isomorphism

$$ \begin{equation} E_1^{p,-(p+s),*}=(N_p/N_{p-1})\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk). \end{equation} \tag{11} $$
Therefore, the differential $d_1$ can be regarded as a map
$$ \begin{equation} d_1\colon(N_p/N_{p-1})\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk)\to (N_{p-1}/N_{p-2})\otimes \operatorname{Ext}_A^{s+1,*}(\Bbbk,\Bbbk). \end{equation} \tag{12} $$

The definition of the Buchstaber filtration implies that

$$ \begin{equation*} \operatorname{im} \widehat\Delta\subset N_{p-1}\otimes A_*, \quad\text{where}\ \ \widehat\Delta=\Delta - 1\otimes \operatorname{id} - \operatorname{id}\mathrel{\otimes} 1\colon N_p\to A_*\otimes A_*. \end{equation*} \notag $$
Therefore, $\widehat\Delta$ induces a map
$$ \begin{equation} \begin{gathered} \, D \colon N_p/N_{p-1} \to N_{p-1}/N_{p-2}\otimes A_*, \\ (x\text{ mod } N_{p-1}) \mapsto \sum_j(-1)^{|x^{(1)}_j|}(x^{(1)}_j\text{ mod } N_{p-2})\otimes x^{(2)}_j. \end{gathered} \end{equation} \tag{13} $$

Theorem 2. (a) The image of $D\colon N_p/N_{p-1}\to N_{p-1}/N_{p-2}\otimes A_*$ is contained in $N_{p-1}/N_{p-2}\otimes P(A_*)$.

(b) The differential

$$ \begin{equation*} d_1\colon(N_p/N_{p-1})\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk)\to (N_{p-1}/N_{p-2})\otimes \operatorname{Ext}_A^{s+1,*}(\Bbbk,\Bbbk) \end{equation*} \notag $$
is decomposed into the composition
$$ \begin{equation*} \begin{aligned} \, (N_p/N_{p-1})\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk) &\xrightarrow{D\otimes 1}(N_{p-1}/N_{p-2})\otimes P(A_*)\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk) \\ &\xrightarrow{1\otimes \delta\otimes 1}(N_{p-1}/N_{p-2})\otimes \operatorname{Ext}_A^{1,*}(\Bbbk,\Bbbk)\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk) \\ &\xrightarrow{1\otimes m}(N_{p-1}/N_{p-2})\otimes \operatorname{Ext}_A^{s+1,*}(\Bbbk,\Bbbk), \end{aligned} \end{equation*} \notag $$
where $m$ is the multiplication in $\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$, which is defined in terms of the standard bar resolution $B(\Bbbk,A_*,\Bbbk)$ [63].

(c) The differential

$$ \begin{equation*} d_1\colon N_p/N_{p-1}\to (N_{p-1}/N_{p-2})\otimes \operatorname{Ext}_A^{1,*}(\Bbbk,\Bbbk) \end{equation*} \notag $$
is injective.

Proof. Consider an arbitrary element
$$ \begin{equation*} a\otimes b \in (N_p/N_{p-1})\otimes \operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk) \end{equation*} \notag $$
and its representative
$$ \begin{equation*} x\otimes y\in N_p\otimes \bar{A}_*^{\otimes s}\subset N_p\otimes F(\Bbbk, A_*,\Bbbk). \end{equation*} \notag $$
Note that $y$ is a cycle in the cobar construction $F(\Bbbk, A_*,\Bbbk)$. Then the element $d_1(a\otimes b)$ is represented by the following cochain in $(N_{p-1}/N_{p-2})\otimes \operatorname{Ext}_A^{s+1,*}(\Bbbk,\Bbbk)$:
$$ \begin{equation*} 1\otimes x\otimes y + (-1)^{| x^{(1)}_j|}x^{(1)}_j\otimes x^{(2)}_j\otimes y\in F^{s+1}(N_{p-1}, A_*,\Bbbk). \end{equation*} \notag $$

Now we need to drop elements of filtration degree less than $p-1$ in the last expression. To do this, we construct an additive basis in $\bar{A}_*$ (for simplicity, we can assume that its elements are homogeneous) recursively: first, we choose a basis in $N_1$, then we complement it to a basis of $N_2$, and so on.

Then $\widehat \Delta x$ can be written as $\displaystyle\sum_i e_i\otimes z_i$, where the elements $e_i$ are pairwise distinct and belong to the chosen basis, and $z_i\in \bar{A}_*$. By the construction of the filtration, all elements $e_i$ belong to $N_{p-1}$. Among them we choose the elements $e_{i_k}$ that do not belong to $N_{p-2}$. Then

$$ \begin{equation*} D (x \text{ mod } N_{p-1})=\sum_k e_{i_k}\otimes z_{i_k}\in (N_{p-1}/N_{p-2}) \otimes \bar{A}_*. \end{equation*} \notag $$
Now it is easy to see that $d_1(a\otimes b)$ is represented by the cochain
$$ \begin{equation*} \sum_k e_{i_k}\otimes z_{i_k}\otimes y \in N_{p-1}\otimes \bar{A}_*^{\otimes s+1}\subset F(N_{p-1},\bar{A}_*,\Bbbk) \mod F(N_{p-2},\bar{A}_*,\Bbbk). \end{equation*} \notag $$
This cochain must be a cocycle in the corresponding quotient complex, whence we obtain
$$ \begin{equation*} \sum_k e_{i_k}\otimes \widehat\Delta(z_{i_k})\otimes y = 0\mod F(N_{p-2},\bar{A}_*,\Bbbk). \end{equation*} \notag $$
Since the elements $e_{i_k}$ are linearly independent, after projecting the above expression onto $N_{p-1}/N_{p-2}$ we obtain that all the $\widehat \Delta z_{i_k}$ map to zero, and therefore all elements $z_{i_k}$ are primitive in $A_*$. This immediately implies statements (a) and (b).

To prove statement (c), consider an element $a\in N_p/N_{p-1}$ represented by some $x\in N_p$. Then $d_1(a)$ is represented by the class of the element

$$ \begin{equation*} \displaystyle\sum_k e_{i_k}\otimes z_{i_k}\in N_{p-1}/N_{p-1}\otimes P(A_*). \end{equation*} \notag $$
The projections of the elements $e_{i_k}$ onto $N_{p-1}/N_{p-2}$ are linearly independent, so if the sum above is zero, then all the $z_{i_k}$ are zero. Thus, in the sum $\widehat\Delta(x)=\displaystyle\sum e_i\otimes z_i$ all elements $e_i$ belong to $N_{p-2}$, so that $x\in N_{p-1}$ and $a=0$. $\Box$

3.5. The $\operatorname{Bss}$-operations in the cohomology of Hopf algebras

The suspension $sV$ of a graded vector space $V=\bigoplus\limits_{j\in \mathbb{Z}} V^{j}$ is the graded vector space $\bigoplus\limits_{j\in \mathbb{Z}}(sV)^{j}$, where $(sV)^j=V^{j-1}$. We denote the graded vector space $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$ by $H=H^*$.

The tensor algebra $T(V)=\bigoplus\limits_{k\geqslant 0} V^{\otimes k}$ of a graded vector space $V$ is equipped with the so-called shuffle product $ \text{ш} $, transforming $T(sH)$ into an associative and graded commutative algebra. Recall that

$$ \begin{equation*} (x_1\otimes \cdots \otimes x_p) \mathbin{ \text{ш} }(x_{p+1}\otimes \cdots \otimes x_{p+q})=\sum_{\sigma\in S_{p,q}}(-1)^{\varepsilon} x_{\sigma(1)} \otimes x_{\sigma(2)}\otimes \cdots\otimes x_{\sigma(p+q)}, \end{equation*} \notag $$
where $S_{p,q}$ is the set of $(p,q)$-shuffles, that is, permutations $\sigma\in S_{p+q}$ satisfying
$$ \begin{equation*} \sigma(1)<\cdots < \sigma(p)\quad\text{and}\quad \sigma(p+1)<\cdots <\sigma (p+q), \end{equation*} \notag $$
and $\varepsilon=\displaystyle\sum_{(i-j)(\sigma(i)-\sigma(j))<0}|x_i|\cdot |x_j|$.

The shuffle product can be defined recursively:

(1) $x\mathbin{ \text{ш} } 1=1\mathbin{ \text{ш} } x=x$ for any $x\in V$;

(2) $(A\otimes x)\mathbin{ \text{ш} } (B\otimes y)=(-1)^{|x|\cdot (|B| +|y|)} \bigl(A\mathbin{ \text{ш} } (B\otimes y )\bigr) \otimes x+ \bigl((A\otimes x) \mathbin{ \text{ш} } B\bigr)\otimes y$ for any $A,B\in T(V) $ and $x,y\in V$.

Let $\operatorname{gr}(A_*)=\bigoplus\limits_p N_p/N_{p-1}$ be the graded algebra associated with the filtration $\{N_p\}$ of $A_*$.

Using the isomorphism $\delta\colon P(A_*)\to \operatorname{Ext}_A^{1,*}(R,R)$ we may think of the homomorphism $D\colon N_p/N_{p-1}\to N_{p-1}/N_{p-2}\otimes P(A_*)$ as a homomorphism $N_p/N_{p-1}\to N_{p-1}/N_{p-2}\otimes sH$ preserving the grading. We define the homomorphism

$$ \begin{equation} \begin{aligned} \, \widetilde{D}\colon N_p/N_{p-1} &\to N_{p-1}/N_{p-2}\otimes sH, \\ (x\text{ mod } N_{p-1}) &\mapsto \sum_j\bigl(x^{(1)}_j \text{ mod } N_{p-2}\bigr)\otimes x^{(2)}_j, \end{aligned} \end{equation} \tag{14} $$
which differs from $D$ by a sign in some graded components (see also formula (13)).

Theorem 3. (i) The iterated homomorphism $\widetilde{D}$, that is,

$$ \begin{equation*} \begin{gathered} \, \widetilde D_p=(\widetilde{D}\otimes 1\otimes\cdots\otimes 1) \circ \cdots \circ (\widetilde{D}\otimes 1)\circ \widetilde{D}, \\ \widetilde D_p\colon N_p/N_{p-1}\to N_{p-1}/N_{p-2}\otimes sH \to N_{p-2}/N_{p-3}\otimes (sH)^{\otimes 2}\to \cdots \to (sH)^{\otimes p}, \end{gathered} \end{equation*} \notag $$
is injective.

(ii) The sum of all iterations of $\widetilde{D}$ is an injective algebra homomorphism

$$ \begin{equation*} \begin{aligned} \, \widehat{D}&=\bigoplus \widetilde D_p\colon \operatorname{gr} (A_*) \to (T(V),\mathbin{ \text{ш} }), \\ \widehat{D}&=\bigoplus \widetilde D_p\colon \bigoplus_p N_p/N_{p-1} \to \bigoplus_p (sH)^{\otimes p}, \end{aligned} \end{equation*} \notag $$
where $ \bigoplus\limits_p N_p/N_{p-1}$ is identified with $\operatorname{gr}(A_*)$ and $\bigoplus\limits_p (sH)^{\otimes p}$ is $T(V)$ with the $\mathbin{ \text{ш} }$-product.

Proof. The proof of (i) literally repeats the proof of statement (c) of Theorem 2.

To prove statement (ii) we use the inductive description of the $\mathbin{ \text{ш} }$-product. If $x,y\in N_1$ and $x,y\notin N_0$, then $\widehat D(x)=x $, $\widehat D(y) =y$, $\widehat D(1)=1$, and it is easy to verify that

$$ \begin{equation*} \widehat D(x)\mathbin{ \text{ш} } 1=1 \mathbin{ \text{ш} } \widehat D(x)=\widehat D(x) \end{equation*} \notag $$
and
$$ \begin{equation*} \widehat D(xy)=x\otimes y+(-1)^{|x|\cdot|y|} y\otimes x= \widehat D(x)\mathbin{ \text{ш} } \widehat D(y). \end{equation*} \notag $$

Now let $x\in N_p/N_{p-1}$ and $y\in N_q/N_{q-1}$. We use induction on $p+q$. For $p+q\leqslant 1$ the statement is true, as we have just seen.

Let

$$ \begin{equation*} \widetilde D (x)=\sum_i x^{(1)}_i\otimes x^{(2)}_i\quad\text{and}\quad \widetilde D (y)=\sum_j y^{(1)}_j\otimes y^{(2)}_j, \end{equation*} \notag $$
where
$$ \begin{equation*} x_i^{(1)}\in N_{p-1}/N_{p-2},\quad y_j^{(1)}\in N_{q-1}/N_{q-2}\quad\text{and}\quad x^{(2)}_i,y^{(2)}_j\in P(A_*)=sH. \end{equation*} \notag $$
The definition of $\widehat D$ implies that
$$ \begin{equation*} \widehat D (x)=\sum_i \widehat D(x^{(1)}_i)\otimes x^{(2)}_i\quad\text{and}\quad \widehat D (y)=\sum_j \widehat D(y^{(1)}_j)\otimes y^{(2)}_j. \end{equation*} \notag $$
Therefore,
$$ \begin{equation*} \begin{aligned} \, \widehat D(xy)&=\widetilde D_{p+q}(xy)=\sum_j\widetilde D_{p+q-1}(xy_j^{(1)}) \otimes y_j^{(2)}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad+\sum_i (-1)^{|x_i^{(2)}|\cdot |y|}D_{p+q-1}(x_i^{(1)}y) \otimes x_i^{(2)} \\ &=\sum_j \widehat D(xy_j^{(1)})\otimes y_j^{(2)}+\sum_i (-1)^{ |x_i^{(2)}| \cdot |y|} \widehat D(x_i^{(1)}y)\otimes x_i^{(2)}. \end{aligned} \end{equation*} \notag $$
By the induction hypothesis this is equal to
$$ \begin{equation*} \sum_j (\widehat D(x)\mathbin{ \text{ш} } \widehat D(y_j^{(1)}))\otimes y_j^{(2)}+ \sum_i (-1)^{|x_i^{(2)}|\cdot |y|}( \widehat D(x_i^{(1)})\mathbin{ \text{ш} } \widehat D( y))\otimes x_i^{(2)}, \end{equation*} \notag $$
which is precisely $\widehat D(x)\mathbin{ \text{ш} } \widehat D(y)$. $\Box$

Differentials of $\operatorname{Bss}$ define some partial multivalued operations $\operatorname{Bss}_p$ on $\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$, which we describe next.

Consider the differential $d_p\colon E_p^{p,-s-p+1,*}\to E^{0,-s,*}_p$ hitting the zero column. By the definition of the filtration $\Phi^p\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$, the image of $d_p$ coincides with the quotient

$$ \begin{equation*} \Phi^p\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)/\Phi^{p-1} \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk) \subset \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)/\Phi^{p-1} \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)\subset E^{0,-s,*}_p. \end{equation*} \notag $$
Therefore, the differential $d_p$ can be thought of as a multivalued map to $\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$, defined up to adding elements of the subgroup $\Phi^{p-1}\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$.

The domain of the differential $d_p$ is the group $E_p^{p,-s-p+1,*}$, which is a subgroup of the quotient $E_1^{p,-s-p+1}/K_{p,s}$, where the subgroup $K_{p,s}$ is defined by the images of the differentials $d_1,\dots,d_{p-1}$ in the groups $E_r^{p,-s-p+1,*}$.

Then the operation $\operatorname{Bss}_p$ is defined by the following commutative diagram, in which the unsigned arrows are obvious inclusions or projections:

It is clear from the diagram that the operation $\operatorname{Bss}_p$ is defined on a subgroup of $(sH)^{\otimes p}\otimes \operatorname{Ext}^{s-1,*}_A(\Bbbk,\Bbbk)$, and its values are subsets of $\Phi^p\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$, which are cosets with respect to the subgroup $\Phi^{p-1}\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$. If the index $p$ is clear from the context, then we refer to the operation $\operatorname{Bss}_p$ as a $\operatorname{Bss}$-operation. For example, the operation $\operatorname{Bss}_1$ is defined on the whole of $sH\otimes \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ and is single-valued. It is easy to see that it coincides up to a sign with multiplication or, equivalently, with the two-fold Massey operation (see § 6.1).

3.6. Realisation of classes in $\operatorname{Ext}_A^{s,*}(\Bbbk,\Bbbk)$ by $\operatorname{Bss}$-operations on the tensor algebra $T(s\operatorname{Ext}_A^{1,*}(\Bbbk,\Bbbk))$

We say that an element $x\in \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ is realised by a non-trivial operation $\operatorname{Bss}_p$ if there exists $y\in (sH)^{\otimes p}\otimes \operatorname{Ext}^{s-1,*}_A(\Bbbk,\Bbbk)$ such that $x\in \operatorname{Bss}_p(y)$ and $0\notin \operatorname{Bss}_p(y)$. We say that $x$ is realised by a unique $\operatorname{Bss}$-operation if $\operatorname{Bss}_p(y_1)=\operatorname{Bss}_p(y_2)$ whenever $x\in \operatorname{Bss}_p(y_1)$ and $x\in \operatorname{Bss}_p(y_2)$.

Theorem 4. Any cohomology class $x\in \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ is realised by a unique non-trivial $\operatorname{Bss}$-operation.

Proof. Recall that the images of differentials of the Buchstaber spectral sequence define an increasing exhaustive filtration $\Phi^p\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$ in the cohomology of the algebra $A$. Suppose that $x\in \Phi^p\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$, but $x\notin\Phi^{p-1} \operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$. Then by the construction of the operation $\operatorname{Bss}_p$ there exists $y\in (sH)^{\otimes p}\otimes \operatorname{Ext}^{s-1,*}_A(\Bbbk,\Bbbk)$ such that $x\in \operatorname{Bss}_p(y)$. Furthermore, since $x\notin\Phi^{p-1} \operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$, the element $x$, or more precisely the classes that it defines in the groups $E_r^{0,-s,*}$, do not lie in the images of the differentials $d_1,\dots,d_{p-1}$. Therefore, $0\notin \operatorname{Bss}(y)$.

Uniqueness of realisation follows from the fact that two cosets with respect to one subgroup either do not intersect or coincide. $\Box$

In other words, the images of the $\operatorname{Bss}$-operations divide each group $\operatorname{Ext}_A^{*,*}(\Bbbk,\Bbbk)$ into pairwise disjoint cosets by the subgroups $\Phi^p\operatorname{Ext}_A^{*,*}(\Bbbk,\Bbbk)$, which do not contain zero.

Corollary 2. Any class $x\in \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ is realised by iterated $\operatorname{Bss}$ operations over classes from $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$.

3.7. Lie algebras

Let $\mathfrak{g}$ be a Lie algebra over a field $\Bbbk$. Note that the well-known subtleties occurring in the case of field characteristic 2 or 3 do not affect the further arguments. The universal enveloping algebra $U\mathfrak{g}$ is a Hopf algebra. Recall that the cohomology $H^*(\mathfrak{g},N)$ of a Lie algebra $\mathfrak{g}$ with coefficients in a $\mathfrak{g}$-module $N$ is $\operatorname{Ext}^{*}_{U\mathfrak{g}}(\Bbbk,N)$. In particular, the cohomology $H^*(\mathfrak{g},(U\mathfrak{g})^*)$ is trivial. The constructions and results presented above in this section are applicable to the cohomology of the Lie algebra $\mathfrak{g}$.

At the same time there are some subtleties here. Namely, in the general case the Buchstaber filtration

$$ \begin{equation*} \Bbbk=N_0\subset N_1 \subset \cdots \subset N_p \subset \cdots \subset(U\mathfrak{g})^* \end{equation*} \notag $$
does not necessarily exhaust $(U\mathfrak{g})^*$. In fact, by Lemma 3 the condition $\bigcup\limits_p N_p (U\mathfrak{g})^*$ is equivalent to $\bigcap\limits_p I^p =0$, where $I$ is the augmentation ideal of $U\mathfrak{g}$. The condition $\bigcap\limits_p I^p =0$ is equivalent to the residual nilpotency of the Lie algebra $\mathfrak{g}$ (see [65], Corollary 3.5). Recall that for a Lie algebra $\mathfrak{g}$ the lower central series is given by
$$ \begin{equation*} \mathfrak{g}\supset \gamma_2(\mathfrak{g})=[\mathfrak{g},\mathfrak{g}] \supset\cdots\supset \gamma_p(\mathfrak{g})\supset\cdots,\quad\text{where}\ \ \gamma_{p}(\mathfrak{g})=[\mathfrak{g},\gamma_{p-1}(\mathfrak{g}_n)]. \end{equation*} \notag $$
A Lie algebra is called nilpotent if there exists $p$ such that $\gamma_{p}(\mathfrak{g})= 0$, and residually nilpotent if $\bigcap\limits_p \gamma_{p}(\mathfrak{g})=0$. A finite-dimensional residually nilpotent Lie algebra is nilpotent.

It follows that for residually nilpotent Lie algebras the module $N_0=\Bbbk\subset (U\mathfrak{g})^*$ is complete, and so the Buchstaber spectral sequence is well defined and converges to the trivial module $E_\infty^{*,*}$, but it is bigraded rather than trigraded. All properties proved above are satisfied for the bigraded $\operatorname{Bss}$. If the Lie algebra $\mathfrak{g}$ is graded by positive integers, then the grading is carried over to $U\mathfrak{g}$ and $(U\mathfrak{g})^*$, and then $\operatorname{Bss}$ becomes trigraded. These Lie algebras will be considered in §§ 7 and 8.

In the case of Lie algebras, instead of the bar complex computing $\operatorname{Ext}^*_{U\mathfrak{g}}(\Bbbk,-)$, it is convenient to use the Chevalley–Eilenberg resolution of the trivial module $\Bbbk$. The latter is the complex $C_*(\mathfrak{g})$ with differential $d_{\rm CE}$, where

$$ \begin{equation*} C_s(\mathfrak{g})=U\mathfrak{g}\otimes \Lambda^s \mathfrak{g}, \end{equation*} \notag $$
and the differential $d_{\rm CE}\colon C_s(\mathfrak{g})\to C_{s-1}(\mathfrak{g})$ is given by
$$ \begin{equation} \begin{aligned} \, \notag &d_{\rm CE} (u\otimes g_1\wedge\cdots \wedge g_s)= \sum_{i=1}^s (-1)^{i+1} u g_{i} \otimes g_1\wedge\cdots \wedge\widehat{g}_i\wedge\cdots \wedge g_s \\ &\qquad+\sum_{i,j} (-1)^{i+j} u\otimes [g_i,g_j]\wedge g_1\wedge\cdots\wedge \widehat{g}_i\wedge\cdots \wedge\widehat{g}_j\wedge\cdots \wedge g_s. \end{aligned} \end{equation} \tag{15} $$

For a graded Lie algebra $\mathfrak{g}$ with finite-dimensional homogeneous components (such as, for example, the infinite-dimensional Lie algebra $L_1$) the dual complex has the form $(U\mathfrak{g})^*\otimes \Lambda^s \mathfrak{g}^*$. The Buchstaber filtration in $(U\mathfrak{g})^*$ starting with $N_0=\Bbbk$ defines a trigraded spectral sequence.

Theorem 5. Let $\mathfrak{g}$ be a nilpotent Lie algebra such that $\gamma_n(\mathfrak{g})=0$. Then all differentials $d_r$ with $r\geqslant n$ in the Buchstaber spectral sequence are zero, in particular, $E^{*,*}_{n}=E^{*,*}_\infty$.

Proof. From the standard description of the spectral sequence of a filtered complex (see, for example, [63], Chap. XI) it is clear that if the differential changes the filtration by at most $k$, then the spectral sequence stabilises at the $(k+1)$th term, that is, $E_{k+1}=E_{\infty}$.

Consider the homological Chevalley–Eilenberg complex $C_s(\mathfrak{g})=U\mathfrak{g}\otimes \Lambda^s\mathfrak{g}$ with differential (15). The powers of the augmentation ideal define a filtration $F^pC_s(\mathfrak{g})=I^p \otimes \Lambda^s \mathfrak{g}$ of $C_s(\mathfrak{g})$. We claim that the differential (15) changes the filtration by at most $n-1$. Indeed, an element $x\in F^pC_s(\mathfrak{g})$ can be written as $\displaystyle\sum x_i\otimes y_i,$ where $x_i\in I^p$ and $y_i\in \Lambda^s \mathfrak{g}$. Then $d_{\rm CE} x$ has the form $\displaystyle\sum x_i g_j \otimes {\widetilde y}_{ij}$, where $g_j\in \mathfrak{g}$, ${\widetilde y}_{ij}\in \Lambda^{s+1}\mathfrak{g}$. Corollary 3.3 in [65] implies that $\gamma_j(\mathfrak{g})= \mathfrak{g} \cap I^j$, so $I^n$ does not contain non-zero elements of $\mathfrak{g}$ by assumption. Therefore, all products $x_i g_j$ belong to $I^{k}$ with $k<n+p$. It follows that the differential $d_{\rm CE}$ in the homology complex changes the filtration by at most $n-1$. By Lemma 3 the Buchstaber filtration in $(U\mathfrak{g})^*$ is dual to the filtration by powers of the augmentation ideal, so passing to the dual complex we obtain the required assertion. $\Box$

4. An example: the primitively generated polynomial algebra

Here we consider an illustrative example of the polynomial algebra $A=\Bbbk[x_1,\dots,x_n]$, where $|x_i|=2$ and $\Bbbk$ is a field. We define the diagonal by the relations $\Delta x_i=1\otimes x_i+x_i\otimes 1$, that is, multiplicative generators are primitive elements. We obtain a Hopf algebra which is isomorphic to the universal enveloping algebra of a real Abelian Lie algebra of dimension $n$, that is, the real Lie algebra with basis $e_i$, $i=1,\dots,n$, and zero commutators $[e_i,e_j]=0$ for all $i$, $j$.

We prove directly that the $\operatorname{Bss}$ of this Hopf algebra stabilises at the term $E_2^{*,*,*}$. In other words, we prove that the differentials $d_r$ are trivial for $r\geqslant 2$ (in the general case this is proved in Theorem 5). Moreover, we will see that $E_1^{*,*,*}$ is a complex dual over $\Bbbk$ to the Koszul resolution, which is split into a direct sum by the additional grading.

The dual algebra $A_*$ is isomorphic to the tensor product of $n$ copies of the algebra of divided powers. More precisely, let $\gamma_{i}^{[k]}\in A_*$ denote the element that is equal to one at the monomial $x_i^{k}$ and to zero at all other monomials. For convenience, let $\gamma_{i}^{[0]}=1$. Then the elements $\gamma_1^{[r_1]}\cdots \gamma_n^{[ r_n]}$ form an additive basis of $A_*$. The coproduct is given by

$$ \begin{equation*} \Delta \gamma_1^{[r_1]}\cdots \gamma_n^{[ r_n]}= \sum_{\substack{r_i=s_i+t_i, \ i=1,\dots,n}} \gamma_1^{[s_1]}\cdots \gamma_n^{[ s_n]}\otimes \gamma_1^{[t_1]}\cdots \gamma_n^{[ t_n]}, \end{equation*} \notag $$
and the product is given by
$$ \begin{equation*} (\gamma_1^{[s_1]}\cdots \gamma_n^{[ s_n]})\cdot (\gamma_1^{[t_1]}\cdots \gamma_n^{[ t_n]})=\biggl(\,\prod_{i=1}^n\frac{(s_i+t_i)!}{s_i!\,t_i!}\biggr) \gamma_1^{[s_1+t_1]}\cdots \gamma_n^{[s_n+t_n]}. \end{equation*} \notag $$
This implies, in particular, that the module of primitives of $A_*$ is additively generated by the elements $\gamma_i^{[1]}$.

We compute the Buchstaber filtration in $A_*$ starting from $N_0=\Bbbk$. This can be done either directly or via the filtration by powers of the augmentation ideal $I$ of $A$. It is easy to see that $I^p$ consists of linear combinations of monomials $x_1^{i_1}\cdots x_n^{i_n}$ such that $\displaystyle\sum_j i_j\geqslant p$. In particular,

$$ \begin{equation*} I^{p}=I^{p+1}\oplus \Bbbk\biggl\langle x_1^{i_1}\cdots x_n^{i_n}\colon \sum_j i_j=p\biggr\rangle. \end{equation*} \notag $$
Then by Lemma 3,
$$ \begin{equation*} N_p=\Bbbk\biggl\langle \gamma_1^{[r_1]}\cdots \gamma_n^{[r_n]}\colon \sum_j r_j\leqslant p \biggr\rangle \end{equation*} \notag $$
and
$$ \begin{equation*} N_p=N_{p-1} \oplus \Bbbk\biggl\langle \gamma_1^{[r_1]}\cdots \gamma_n^{[r_n]}\colon \sum_j r_j= p \biggr\rangle. \end{equation*} \notag $$

To calculate $\operatorname{Ext}^{*,*}_{A}(\Bbbk,\Bbbk)$ we use the Koszul resolution, as it is the most convenient tool in this case. We assign the bigrading $(0,2)$ to the polynomial generators $x_i$. Let $V$ be a vector space with basis $y_1,\dots,y_n$, where the elements $y_i$ have bigrading $(-1,2)$. The Koszul resolution is the bigraded differential algebra $\Bbbk[x_1,\dots,x_n]\otimes\Lambda(y_1,\dots,y_n)$ with differential

$$ \begin{equation} d(f(x_1,\dots,x_n)\otimes y_{k_1}\wedge\cdots \wedge y_{k_s})= \!\sum_{i=1}^s(-1)^{i+1} f(x_1,\dots,x_n) x_{k_i} \otimes y_{k_1}\wedge\cdots \wedge {\widehat y}_{k_i}\wedge\cdots \wedge y_{k_s} \end{equation} \tag{16} $$
of bidegree $(1,0)$. It is easy to see that the resolution consists of free $A$-modules $P^s=A\otimes \Lambda^s V$. Applying the functor $\operatorname{Hom}_A^*(-,\Bbbk)$ to the Koszul resolution, we obtain the complex of modules $\operatorname{Hom}^*_\Bbbk(\Lambda^s V,\Bbbk)$ with trivial differential. Thus, $\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ is isomorphic to the $\Bbbk$-module $(\Lambda^s V)^*$ dual to $\Lambda^s V$.

Now note that the Koszul resolution has the third grading by the weights of the monomials: by definition, a monomial $x_1^{r_1}\cdots x_n^{r_n}\otimes y_{k_1}\wedge\cdots \wedge y_{k_s}$ has weight $s+\displaystyle\sum r_j$. (Informally, the weight of a monomial is the number of the symbols $x_i$ and $y_i$ in the notation for it.) It is easy to see that the differential preserves weights, and the entire Koszul resolution decomposes into a direct sum of acyclic subcomplexes consisting of elements of the same weight.

Theorem 6. The $(-q)$th row of the trigraded spectral sequence

$$ \begin{equation*} E^{0,-q,*}_1\xleftarrow{d_1}E^{1,-q,*}_1\xleftarrow{d_1} E^{2,-q,*}_1\xleftarrow{d_1}\cdots \end{equation*} \notag $$
is isomorphic to the dual complex of the direct summand of the Koszul resolution consisting of elements of weight $q$. In particular, each row is exact, and therefore $E^{*,*,*}_2=E^{*,*,*}_\infty$.

Proof. We have an isomorphism
$$ \begin{equation*} E_1^{p,-q,*}=(N_p/N_{p-1})\otimes \operatorname{Ext}^{q-p,*}_A(\Bbbk,\Bbbk)= (N_p/N_{p-1})\otimes (\Lambda^{q-p} V)^*. \end{equation*} \notag $$
We use Theorem 2 to calculate
$$ \begin{equation*} d_1\colon (N_p/N_{p-1})\otimes (\Lambda^{q-p} V)^*\to (N_{p-1}/N_{p-2})\otimes (\Lambda^{q-p+1} V)^*. \end{equation*} \notag $$
The formula for the coproduct in $A_*$ implies that the map $D\colon (N_p/N_{p-1})\to (N_{p-1}/N_{p-2})\otimes P(A_*)$ is given by
$$ \begin{equation*} D(\gamma_1^{[r_1]}\cdots \gamma_n^{[ r_n]})=\sum_{j=1}^n\gamma_1^{[r_1]} \cdots \gamma_j^{[ r_j-1]}\cdots \gamma_n^{[ r_n]}\otimes \gamma_j^{[1]}, \end{equation*} \notag $$
where $\displaystyle\sum r_i=p$, and if $r_j=0$ for some $j$ then the corresponding summand on the right hand side is omitted. Therefore,
$$ \begin{equation*} d_1(\gamma_1^{[r_1]}\cdots \gamma_n^{[ r_n]}\otimes y^*_{i_1}\wedge\cdots \wedge y^*_{i_{q-p}})=\sum_{j=1}^n\gamma_1^{[r_1]}\cdots \gamma_j^{[ r_j-1]} \cdots \gamma_n^{[ r_n]}\otimes y_j^*\wedge y^*_{i_1}\wedge\cdots \wedge y^*_{i_{q-p}}. \end{equation*} \notag $$
It is easy to see that the same formula for $d_1$ is obtained from formula (16) for the differential in the Koszul resolution by passing to dual modules, and it contains only elements of weight $q$. $\Box$

For this example the filtration $\Phi^r$ is easy to describe. Indeed, $d_1\colon E_1^{1,-q,*}\to E_1^{0,-q,*}$ is an epimorphism, so

$$ \begin{equation*} \Phi^1=\operatorname{Ext}^{*,*}_{\Bbbk[x_1,\dots,x_n]}(\Bbbk,\Bbbk). \end{equation*} \notag $$
Hence $\Phi^r$ also coincides with $\operatorname{Ext}^{*,*}_{\Bbbk[x_1,\dots,x_n]}(\Bbbk,\Bbbk)$ for $r\geqslant 2$, that is, $\operatorname{Bss}$ degenerates in the second term.

5. Calculating filtrations

5.1. The Buchstaber filtration in $S_*\otimes \mathbb{Q}=\Omega_U^*\otimes \mathbb{Q}$

We recall some basic facts from the theory of $U$-cobordism. The coefficient ring $\Omega_U^*=U^*(\operatorname{pt})$ is the polynomial ring $\Omega_U^*=\mathbb{Z}[y_1,y_2,\ldots]$ in generators $y_i$, $|y_i|=-2i$, for $i\geqslant 1$. The algebra of stable cohomology operations in the theory of $U$-cobordism is the completed tensor product of the coefficient ring $\Omega_U^*$ and the Landweber–Novikov algebra $S$ (see [29] and [66]). An additive basis of $S$ is formed by the operations $S_\omega$ indexed by the partitions $\omega=(i_1,\dots,i_k)$ of positive integers $n$ into sums of positive integers $\displaystyle\sum i_k=n$, $|S_\omega|=2n$. It is convenient to assume that the unity of the algebra $S$ corresponds to the ‘empty’ partition. Note that partitions that differ only in the order of their summands are considered to be identical. Comultiplication in the algebra $S$ is defined by

$$ \begin{equation*} \Delta S_\omega=\sum_{\omega_1\sqcup \omega_2=\omega} S_{\omega_1}\otimes S_{\omega_2}. \end{equation*} \notag $$

The fact that $S$ is an algebra (that is, $S$ is closed under multiplication) was proved in [29] and [66]. Note that the formulae for the product in $S$ are not yet sufficient to describe multiplication in the algebra of stable operations in the theory of $U$-cobordism. Indeed, to represent the composition of two elements of $\Omega_U^* \mathbin{\widehat\otimes} S$ as an element of this tensor product, one also needs to know the commutation relations for elements of $\Omega_U^*$ and $S$. The corresponding formulae were obtained in [29].

Next, we compute the Buchstaber filtration in the algebra $S_*$ by using the remarkable isomorphism $S_*\otimes \mathbb{Q}= \Omega_U^*\otimes \mathbb{Q}$ (see [25] and [34]).

There is an injective morphism of $S$-modules $\phi\colon \Omega_U^*\to S_*$ given by

$$ \begin{equation*} \phi_x(s)=\mu(s(x)), \end{equation*} \notag $$
where $\mu\colon\Omega_U^*\to \mathbb{Z}$ is the augmentation, $s\in S$, and $x\in \Omega_U^*$; see [25]. Moreover, it is easy to check that $\phi$ is compatible with the ring structures. Finally, it follows from [25], Lemma 2.1, and [34], Lemma 2.2, that $\phi\otimes\mathbb{Q}$ is an isomorphism. Thus, to construct a trigraded spectral sequence, instead of the $(S\otimes \mathbb{Q})$-module $S_*\otimes \mathbb{Q}$, we can consider $\Omega_U^*\otimes\mathbb{Q}$.

We describe the Buchstaber filtration starting from $N_0=\mathbb{Q}\subset\Omega_U^*\otimes \mathbb{Q}$. To do this we need convenient polynomial generators in $\Omega_U^*\otimes\mathbb{Q}$ and a description of the action of $S$ in terms of these generators.

The Chern–Dold character was defined in [25] as a natural transformation of cohomology theories

$$ \begin{equation*} \operatorname{ch}_U\colon U^*(X)\to H^*(X,\Omega_U^*\otimes\mathbb{Q}). \end{equation*} \notag $$
For $X=\operatorname{pt}$, the Chern–Dold character is the natural inclusion $\Omega_U^* \to\Omega_U^*\otimes\mathbb{Q}$. Let $u\in U^2(\mathbb{C} P^\infty)$ be the first $U$-cobordism Chern class of the universal line bundle. Then $\operatorname{ch}_U(u)\in H^*(\mathbb{C} P^\infty, \Omega_U^*\otimes \mathbb{Q})$ is given by the series
$$ \begin{equation*} \operatorname{ch}_U(u)=\beta(z)=z+\sum_{n=1}^\infty t_n \frac{z^{n+1}}{(n+1)!}\,, \end{equation*} \notag $$
where $z\in H^2(\mathbb{C} P^\infty,\mathbb{Z})$ is the standard Chern class in the cohomology of the same universal line bundle. The classes $t_n\in \Omega^{-2n}_U$ were completely characterised in [25]. The existence of representatives for these classes that are smooth irreducible algebraic varieties was an open question for a long time. Recently, a positive answer was obtained in [67], namely, it was shown that the classes $t_n$ are represented by the non-singular theta divisors $\Theta^n$, $n=1,2,\ldots$ , of general principally polarised Abelian varieties. Then $\Omega_U^*\otimes \mathbb{Q}=\mathbb{Q}[t_1,t_2,\ldots]$.

We introduce the formal series

$$ \begin{equation*} Q_v(z)=1+\sum_{n=1}^\infty (-1)^n v_n\frac{z^n}{(n+1)!}\,, \end{equation*} \notag $$
which satisfies
$$ \begin{equation*} Q_v(z)\beta(z)=z. \end{equation*} \notag $$
The last relation can be written in more detail as follows:
$$ \begin{equation} \biggl(1+\sum_{n=1}^\infty (-1)^n v_n \frac{z^n}{(n+1)!}\biggr) \biggl(1+\sum_{n=1}^\infty t_n \frac{z^n}{(n+1)!}\biggr)=1. \end{equation} \tag{17} $$

It is easy to see that the elements $v_1,v_2,\ldots$ are polynomial generators of the ring $\Omega_U^*\otimes \mathbb{Q}$:

$$ \begin{equation*} \Omega_U^*\otimes \mathbb{Q} =\mathbb{Q}[t_1,t_2,\ldots]= \mathbb{Q}[v_1,v_2,\ldots]. \end{equation*} \notag $$
We refer to the basis of $\Omega_U^*\otimes \mathbb{Q}$ consisting of monomials in the generators $v_i$ as the basis of $v$-monomials.

In small dimensions, the generators $t_n$ and $v_n$ are related by the formulae

$$ \begin{equation*} \begin{aligned} \, t_1&=v_1, \\ t_2&=-v_2 + \frac{3}{2}v_1^2, \\ t_3&=v_3 - 4 v_2v_1 + 3 v_1^3, \\ t_4&=- v_4 + 5 v_3 v_1 + \frac{10}{3}v_2^2 - 15 v_2 v_1^2 + \frac{15}{2} v_1^4, \end{aligned} \end{equation*} \notag $$
and
$$ \begin{equation*} t_5=v_5 - 6 v_4v_1 -10 v_3v_2 + \frac{45}{2}v_3 v_1^2 + 30 v_2^2 v_1 -60 v_2 v_1^3 + \frac{45}{2}v_1^5. \end{equation*} \notag $$

In general, the following lemma holds, which is easily proved by induction.

Lemma 5. The element $t_n$ is a polynomial with rational coefficients in the generators $v_i$, $i=1,\dots,n$, in which the coefficients of $v_n$ and $v_1^n$ are non-zero. More precisely, the coefficient of $v_n$ is $(-1)^{n+1}$, and the coefficient of $v_1^n$ is $(n+1)!/2^n$.

The action of the Landweber–Novikov algebra on the generators $t_n$ and $v_n$ is described as follows.

Statement 1 ([34], [67]). (a) The equality $S_\omega t_n=0$ if $\omega\ne (k)$ holds.

(b) $S_{(k)}t_n$ is equal to the coefficient of $z^{n+1}$ in the expansion of $(n+1)!\,\beta(z)^{k+1}$; in particular,

$$ \begin{equation*} \begin{aligned} \, S_{(n)} t_n &=(n+1)!, \\ S_{(n-1)} t_n &=\frac{n(n+1)}{2}\, n!\, t_1, \\ S_{(1)} t_n &=\sum_{k=0}^{n-1} \begin{pmatrix} n+1 \\ k+1\end{pmatrix} t_k t_{n-k-1}. \end{aligned} \end{equation*} \notag $$

(c) $S_{(k)}v_n$ is equal to the coefficient of $z^{n+1}$ in the expansion of $-(n+1)!\,z\beta(z)^{k-1}$; in particular,

$$ \begin{equation*} \begin{aligned} \, S_{(1)} v_1&=-2, \\ S_{(1)} v_n &=0\quad\text{for}\ \ n>1, \\ S_{(2)} v_n &=(-1)^{n+1} n(n+1) t_{n-2}. \end{aligned} \end{equation*} \notag $$

The formulae in (b) and (c) can be written as equalities of formal power series:

$$ \begin{equation*} S_{(k)}(\beta(z))=\beta(z)^{k+1} \quad\text{and}\quad S_{(k)}(Q_v(z))=-z \beta(z)^{k-1}. \end{equation*} \notag $$

Now we proceed to the calculation of the filtration $N_0\subset N_1 \subset\cdots\subset\Omega_U^*\otimes\mathbb{Q}$. Recall that $N_0=\mathbb{Q}\subset \Omega_U^*\otimes\mathbb{Q}$ and

$$ \begin{equation*} N_{p+1}=\{x \in \Omega_U^*\otimes \mathbb{Q}\colon s\cdot x\in N_p \text{ for all } S_\omega, |\omega|>0\}. \end{equation*} \notag $$

It is clear that a homogeneous element $\Omega_U^*\otimes \mathbb{Q}$ of degree $-2d$ lies in $N_d$, but it can also lie in $N_p$ with $p<d$.

It is convenient to encode the monomials $v_1^{i_1}v_2^{i_2}\cdots v_m^{i_m}$ by infinite sequences of non-negative integers $(i_1,i_2,\ldots)$ with a finite number of non-zero elements or, equivalently, by elements of the direct limit of semigroups $\varinjlim(\mathbb{Z}_{\geqslant 0})^n$. For such a sequence $I=(i_1,i_2,\ldots)$, we denote the monomial $\displaystyle\prod_k v_k^{i_k}$ by $v_I$. For $I=(i_1,i_2,\ldots)$ we set

$$ \begin{equation} p(I)=i_1+\sum_{k}(k-1)i_k. \end{equation} \tag{18} $$
We also let $d(I)=\displaystyle\sum_{k} k i_k$; then $|v_I|=-2d(I)$.

Theorem 7. (a) The element $v_I$ lies in $N_{p(I)}$.

(b) The additive basis $\Omega_U^*\otimes \mathbb{Q}$ is compatible with the filtration $N_{p}$ in the following sense:

$$ \begin{equation*} \mathbb{Q}\langle v_I \colon p(I)=p \rangle \oplus N_{p-1}=N_p. \end{equation*} \notag $$
In particular, $v_I$ does not lie in $N_{p(I)-1}$.

We preface the proof with an auxiliary statement.

Lemma 6. The following relations hold.

(a) $v_1\in N_1$, $v_1^k\in N_k$, $v_1\notin N_{k-1}$.

(b) $v_k\in N_{k-1}$ for $k\geqslant 2$.

(c) $t_k\in N_{k}$ and $t_k\notin N_{k-1}$.

(d) $v_k\notin N_{k-2}$ for $k\geqslant 2$.

Proof. (a) We have $S_{(1)}v_1=-2$, and all other operations $S_\omega\ne 1$ vanish at $v_1$, so that $v_1\in N_1$. Since $|v_1^k|=-2k$, we have $v_1^k\in N_k$. The relation $S_{(1)}v_1^k=-2k v_1^{k-1}$ implies that $v_1^k\notin N_{k-1}$, by induction on $k$.

(b) For any $S_{(\omega)}$ except 1 and $S_{(1)}$ the element $S_{\omega}v_k$ belongs to $N_{k-2}$ for dimensional reasons. By Statement 1 we have $S_{(1)}v_k=0\in N_0\subset N_{k-2}$ for $k\geqslant 2$. Therefore, $v_k\in N_{k-1}$.

(c) It is clear that $t_k\in N_k$, since $|t_k|=-2k$. By Lemma 5

$$ \begin{equation*} t_k=\frac{(k+1)!}{2^k} v_1^k + V, \end{equation*} \notag $$
where $V$ denotes a linear combination of monomials containing at least one factor $v_m$ with $m\geqslant 2$. From dimensional considerations, Lemma 2, and statement (b) we obtain $V\in N_{k-1}$. On the other hand $v_1^k\notin N_{k-1}$, so $t_k\notin N_{k-1}$.

(d) Clearly, $v_2\notin N_0$. Further, for $k\geqslant 3$ the element $v_k$ cannot belong to $N_{k-2}$ since $S_{(2)}v_k=(-1)^{k+1} k(k+1) t_{k-2}\notin N_{k-3}$. $\Box$

Proof of Theorem 7. Part (a) follows trivially from Lemma 2 and parts (a) and (b) of Lemma 6.

To prove part (b), we show that no linear combination of monomials $v_I$ of the same degree with $p(I)=p$ can belong to $N_{p-1}$. Suppose this is false, and take the smallest $p$ for which there exists a non-trivial linear combination

$$ \begin{equation*} x= \sum_{\substack{d(I_k)=d \\ p(I_k)=p}} \lambda_k v_{I_k}\in N_{p-1},\quad\text{where}\ \ \lambda_k\in \mathbb{Q}. \end{equation*} \notag $$
Now consider two cases.

First, suppose that there exists a monomial $v_{I_k}$ containing $v_1$ in a non-zero power. Then $S_{(1)}x\ne 0$, and $S_{(1)}x$ is a linear combination of monomials $v_I$ such that $d(I)=d-1$ and $p(I)=p-1$. Indeed, it is easy to verify that $S_{(1)}v_I=0$ if $i_1=0$ in the sequence $I=(i_1,i_2,\ldots)$. If $i_1>0$, then $S_{(1)}v_I=-2 i_1 v_{J}$, where $J=(i_1-1,i_2,\ldots)$. Then $x\in N_{p-1}$ implies that $S_{(1)}x\in N_{p-2}$, which contradicts the minimality of $p$.

Now suppose that $v_1$ does not occur in any of the monomials $v_{I_k}$. In this case $S_{(1)}x=0$, and this gives us nothing. We show that $S_{(2)}x\ne 0$ and $S_{(2)}x$ contains a term $v_{\widehat I}$ such that $p(\widehat I)=p-1$. Then $x\in N_{p-1}$ implies that $S_{(2)}x\in N_{p-2}$, which contradicts the minimality of $p$.

We introduce a right lexicographic order on the sequences of non-negative integers $I=(i_1,i_2,\ldots)$ with a finite number of non-zero elements. Namely, we let $I>J$ if there is an integer $k>0$ such that $i_k>j_k$ and $i_m=j_m$ for $m>k$.

Lemma 7. (a) Let $v_I=v_k^{i_k}v_{k+1}^{i_{k+1}}\cdots v_m^{i_m}$, where $2\leqslant k\leqslant m$ and $i_k>0$. Each monomial $v_J$ in the expansion

$$ \begin{equation*} S_{(2)}v_I=\displaystyle\sum_J \lambda_J v_J \end{equation*} \notag $$
satisfies $p(J)\leqslant p(I)-2$, except the monomials $v_{I_s}$ such that
$$ \begin{equation} I_s=(i_s,0,\dots,0,i_k,\dots,i_s-1,\dots,i_m,0,\dots),\qquad s=k,\dots,m, \end{equation} \tag{19} $$
which satisfy $p(I_s)=p(I)-1$.

(b) Let $I>J$, where $d(I)=d(J)$ and $p(I)=p(J)$. Let $v_I= v_k^{i_k}v_{k+1}^{i_{k+1}}\cdots v_m^{i_m}$, where $2\leqslant k\leqslant m$ and $i_k>0$. Then $I_{k}>J_t$ for all possible $t$.

This lemma implies that if $v_I=v_k^{i_k}v_{k+1}^{i_{k+1}}\cdots v_m^{i_m}$, where $2\leqslant k\leqslant m$ and $i_k>0$, is the leading monomial in the expansion of $x$ in the $v$-basis, then in the expansion of $S_{(2)}x$ one can take as $v_{\widehat I}$ the monomial corresponding to $I_{k}$; its coefficient will be distinct from zero. $\Box$

Proof of Lemma 7. The operation $S_{(2)}$ is a derivation, that is,
$$ \begin{equation*} S_{(2)}(xy)=S_{(2)}(x)y+x S_{(2)}(y). \end{equation*} \notag $$
This implies the identity
$$ \begin{equation} \begin{aligned} \, \notag S_{(2)}(v_j^i)&=i v_j^{i-1} S_{(2)}v_j=i v_j^{i-1} (-1)^{j+1} j (j+1)t_{j-2} \\ \notag &=i v_j^{i-1} (-1)^{j+1} j (j+1) \frac{(j-1)!}{2^{j-2}} v_1^{j-2} +A \\ &=(-1)^{j+1} i\,\frac{(j+1)!}{2^{j-2}}\,v_j^{i-1} v_1^{j-2} + A, \end{aligned} \end{equation} \tag{20} $$
where $A$ is a linear combination of $v$-monomials of degree $ij - 2$, and each monomial contains at least $i$ factors $v_s$, $s>1$.

In the following calculation we do not present the coefficients explicitly as in the last formula. It will be enough to know that such a coefficient is non-zero; we denote coefficients by the character $\epsilon$ with different indices.

Let $v_I= v_k^{i_k}v_{k+1}^{i_{k+1}}\cdots v_m^{i_m}$, where $2\leqslant k\leqslant m$ and $i_k>0$. Let $d= d(I) $ and $p= p(I)$, and note that the number of factors $v_j$ with $j>1$ in the monomial $v_I$ is exactly $d-p$. Then

$$ \begin{equation} \begin{aligned} \, S_{(2)}(v_I)&=\sum_{j=k}^m v_k^{i_k}\cdots S_{(2)}(v_{j}^{i_j})\cdots v_m^{i_m}=\sum_{j=k}^m \epsilon_j v_1^j v_k^{i_k}\cdots (v_{j}^{i_j-1}) \cdots v_m^{i_m} + A, \end{aligned} \end{equation} \tag{21} $$
where $A$ is a linear combination of $v$-monomials of degree $d-2$ and in each of them the factors $v_j$ with $j>1$ occur at least $d-p$ times. Therefore, for such a monomial $v_J$ the value $p(J)$ is at most $(d-2)-(d-p)=p-2$. This implies, in particular, that $A\in N_{p-2}$.

The remaining monomials in the sum on the right-hand side of (21) correspond to the sequences

$$ \begin{equation} I_s=(i_s,0,\dots,0,i_k,\dots,i_s-1,\dots,i_m,0,\ldots),\qquad s=k,\dots,m. \end{equation} \tag{22} $$
It is easy to see that $d(I_s)=d-2$, $p(I_s)=p-1$. Statement (a) is proved.

Now we prove (b). Let $I=(0,\dots, i_k,\dots, i_m,0,\ldots)$, $J=(0,j_2,\ldots)$, $d(I)=d(J)$ and $I>J$. Since $p(I)=p(J)$, then $\displaystyle\sum_t i_t=\displaystyle\sum_t j_t $.

Since $I>J$, there exists $q$ such that $i_q>j_q$ and $i_r=j_r$ for $r>q$. We claim that $q>k$. Assume by contradiction that $q=k$. Then $d(I)=d(J)$ implies that

$$ \begin{equation} k i_k=k j_k + \sum_{t=2}^{k-1} t j_t, \end{equation} \tag{23} $$
and $p(I)=p(J)$ impies that
$$ \begin{equation} i_k=j_k + \sum_{t=2}^{k-1} j_t. \end{equation} \tag{24} $$
Subtracting from (23) equality (24) times $k$ we obtain
$$ \begin{equation*} 0=\sum_{t=2}^{k-1}(k-t) j_t. \end{equation*} \notag $$
Since the coefficients $k-t$ are positive and $j_t\geqslant 0$, the above equality implies that $j_2=\cdots=j_{k-1}=0$. Hence $I$ coincides with $J$ in all positions except the $k$th, and therefore $d(I)\ne d(J)$, which contradicts the assumption of the lemma.

Hence $q>k$. Then the $q$th element of the sequence

$$ \begin{equation*} I_k=(i_k,0,\dots,0,i_k-1,\dots,i_m,0,\ldots) \end{equation*} \notag $$
is strictly greater than the $q$th element of any sequence of the form $J_t$. $\Box$

Remark 1 (filtration by powers of the augmentation ideal in $S\otimes \mathbb{Q}$). Earlier in this section we encoded $v$-monomials by infinite sequences $(i_1,i_2,\ldots)$ with finitely many non-zero elements (elements of the group $\varinjlim(\mathbb{Z}_{\geqslant 0})^n$). Now it is convenient for us to encode $v$-monomials in another way: by the same partitions that encode Landweber–Novikov operations. Namely, the partitions of a positive integer $n$ into sums $n=j_1+\cdots+j_s$ are in a one-to-one correspondence with the $v$-monomials of degree $-2n$: a partition $\omega=(j_1,\dots,j_s)$ corresponds to the $v$-monomial $v_\omega=v_{j_1}\cdots v_{j_s}$. Let

$$ \begin{equation} p(\omega)=\sum_{k: j_k>1}(j_k-1)+\#\{k \colon j_k=1\} . \end{equation} \tag{25} $$
This definition is compatible with. (18).

Recall that the inclusion of $S$-modules $\phi\colon\Omega_U^*\to S_*$ becomes an isomorphism after tensoring with $\mathbb{Q}$. The isomorphism $\phi\otimes \mathbb{Q}$ carries the filtration $N_p$ to $S_*$.

Let $I$ be the augmentation ideal of $S\otimes \mathbb{Q}$. Put $F^pS\otimes \mathbb{Q}=I^p$.

Recall (see [34]) that the elements $v_\omega$ form the basis of $S_*\otimes\mathbb{Q}$ that is dual to the basis consisting of tangential Landweber–Novikov operations (see [29]). The tangential operation corresponding to $\omega$ is denoted by $\overline S_\omega$.

From Lemma 3 one easily obtains the following description of the filtration of $S\otimes \mathbb{Q}$ by powers of the augmentation ideal:

Theorem 8. The following relations hold.

(a) $\overline S_\omega \in F^{p(\omega)} S\otimes \mathbb{Q}= I^{p(\omega)}$.

(b) $\overline S_\omega \notin F^{p(\omega)+1} S\otimes \mathbb{Q}= I^{p(\omega)+1}$. Furthermore,

$$ \begin{equation*} I^p=\mathbb{Q}\langle \overline S_\omega \colon p( \omega)= p \rangle\oplus I^{p+1}. \end{equation*} \notag $$

5.2. Filtration by powers of the augmentation ideal in $UL_1$

Recall that the Lie algebra $L_1$ is generated over $\mathbb{Q}$ by the elements $e_i$, $i=1,2,\ldots$ , that satisfy the relations $[e_i,e_j]=(j-i) e_{i+j}$. The universal enveloping algebra $UL_1$ is the associative algebra over $\mathbb{Q}$ with unity generated by the elements $e_i$, $i\geqslant 1$, that satisfy the relations $e_ie_j-e_j e_i=(j-i) e_{i+j}$. By the Poincaré–Birkhoff–Witt theorem monomials of the form $e_{i_1}\cdots e_{i_n}$, where $i_1\leqslant i_2\leqslant \cdots\leqslant i_n$, form a basis of the universal enveloping algebra $UL_1$.

In many problems related to the Lie algebra $L_1$, a grading of the algebra $L_1$ is used in which the element $e_i$ has degree $i$. The commutator is homogeneous with respect to this grading, and it induces a grading of the universal enveloping algebra $UL_1$: the monomial $e_{i_1}\cdots e_{i_n}$ has degree $\displaystyle\sum_{j=1}^n i_j$.

The second grading of the Chevalley–Eilenberg complex is introduced in a similar way, namely, an element $\omega_{i_1}\wedge\cdots \wedge \omega_{i_n}$ has bidegree $\biggl(n,\displaystyle\sum_{j=1}^n i_j\biggr)$. The differential of the Chevalley–Eilenberg complex preserves the second grading, and therefore the Chevalley–Eilenberg complex of $L_1$ splits into a direct sum of subcomplexes, which results in a non-trivial splitting of the cohomology of $L_1$.

Theorem 9. Let $I$ be the augmentation ideal in $UL_1$. Then

(a) $e_1\in I^1$, $e_1\notin I^2$.

(b) $e_n\in I^{n-1}$, $e_n\notin I^n$ for $n\geqslant 2$.

(c) Let $1< j_1\leqslant\cdots\leqslant j_n$. An element of the form $e_1^s e_{j_1}\cdots e_{j_n}$ belongs to $I^\nu$, where $\nu=s+\displaystyle\sum_{k=1}^n (j_k-1)$.

Proof. Clearly, $e_1,e_2\in I$. Suppose that $e_1\in I^2$. Then $e_1$ can be written as $\sum a_ib_i$, where $a_i,b_i \in I$. The degree of each of the elements $a_i$ and $b_i$ is at least 1, but then the degree of $e_1$ is at least 2.

The identity $e_n=[e_1,e_{n-1}]/(n-2)$ implies that $e_n\in I^{n-1}$ for $n\geqslant 3$. We claim that $e_n\notin I^n$. Assume for a contradiction that $e_n$ is a linear combination of products of length at least $n$. Since the degree of $e_n$ is $n$, there is only one summand in the linear combination, that is, $e_n=\lambda e_1^n$ with $\lambda\in\mathbb{Q}$. This contradicts the Poincaré–Birkhoff–Witt theorem. $\Box$

We write a basis monomial $e_{i_1}\cdots e_{i_n}$ as $x=e_1^m e_{j_1}\cdots e_{j_k}$, where $1<j_1\leqslant \cdots \leqslant j_k$, and put

$$ \begin{equation*} \nu(x)=m+\displaystyle\sum_{s=1}^k (j_s-1). \end{equation*} \notag $$
Now we can give a complete description of the filtration $F^p UL_1=I^p$.

Theorem 10. In the above notation, $x\in F^{\nu(x)}UL_1 \setminus F^{\nu(x)+1}UL_1$. Moreover, the Poincaré–Birkhoff–Witt basis is compatible with the filtration $F^pUL_1$ in the following sense:

$$ \begin{equation*} \mathbb{Q}\langle x=e_1^m e_{j_1}\cdots e_{j_k}\colon \nu(x)= p \rangle \oplus I^{p+1}=I^p. \end{equation*} \notag $$

Proof. We denote the degree of a monomial $x=e_1^m e_{j_1}\cdots e_{j_k}$ by
$$ \begin{equation*} d(x)=m+\displaystyle\sum_{s=1}^k j_s. \end{equation*} \notag $$
Fix a degree $d$ and a parameter $p\leqslant d$. We prove that no linear combination of the form
$$ \begin{equation} y=\sum_{d(x_j)=d,\,\nu(x_j)=p} \lambda_j x_j, \end{equation} \tag{26} $$
where $x_j$ is a monomial of the Poincaré–Birkhoff–Witt basis and $\lambda_j\in \mathbb{Q}$, can belong to $I^{p+1}$. For a contradiction assume that $y=\displaystyle\sum_s \mu_s y_s$, $\mu_s\in \mathbb{Q}$, and each $y_s$ is a product of basis elements $e_i$, $i\geqslant 1$, of length at least $p+1$.

Let $m$ denote the number of factors $e_1$ in the monomial $y_s$, and let $k$ denote the number of factors $e_i$ with $i\geqslant 2$. Then $m+2k\leqslant d$ and $p+1\leqslant m+k$, which implies that $k\leqslant d-p-1$.

We rewrite the relations $e_ie_j - e_je_i=(j-i)e_{i+j}$ in $UL_1$ as

$$ \begin{equation} e_j e_i=e_ie_j-(j-i) e_{i+j}\quad \text{for}\ \ j>i>1, \end{equation} \tag{27} $$
and
$$ \begin{equation} e_j e_1=e_1e_j-(j-1) e_{j+1}\quad \text{for}\ \ j>1. \end{equation} \tag{28} $$
Then replacing successively the occurrences of $\cdots e_je_i \cdots$, $j>i$, in the monomials $y_s$ by the right-hand sides of the above relations, we reduce $y=\displaystyle\sum_s \mu_s y_s$ in a finite number of steps to a linear combination of monomials of the Poincaré–Birkhoff–Witt basis, that is, we obtain the expression (26).

In the process of this reduction, the number of occurrences of the symbols $e_i$, $i\geqslant 2$ does not increase and therefore does not exceed $d-p-1$. On the other hand the number of occurrences of the symbols $e_i$, $i\geqslant 2$, in each monomial $e_1^m e_{j_1}\cdots e_{j_k}$ in (26) is exactly $k=d(x)-\nu(x)=d-p$. $\Box$

Remark 2 (the cohomology of $L_1$ in dimensions 1 and 2). For the generators $\omega_k$ dual to the generators $e_k\in L_1$, $k\geqslant 1$, the differential in the Chevalley–Eilenberg complex $\Lambda (L_1)^*$ is given by the formula

$$ \begin{equation} d\omega_k=\sum_{i=1}^{[(k-1)/2]}(k-2i)\omega_{k-i}\wedge\omega_i. \end{equation} \tag{29} $$
It is easy to see that the space $H^1(L_1)$ is two-dimensional and is generated by the classes $\omega_1$ and $\omega_2$ dual to $e_1$ and $e_2$.

A more complicated but still elementary check shows that $H^2(L_1)$ is also two-dimensional and generated by the classes $x^2_{-}$ and $x^2_{+}$ represented by the cocycles $\omega_3\wedge\omega_2$ and $\omega_5\wedge\omega_2-3\omega_4\wedge\omega_3$, respectively. The class $x^2_{-}$ represents the triple Massey product $\langle -\omega_1,\omega_2,\omega_2\rangle$. (See § 6.1 for the definitions of Massey products, their defining system, and indeterminacy.) The corresponding defining system is

$$ \begin{equation*} \begin{matrix} -\omega_1 & -\omega_3 & \\ &\hphantom{-}\omega_2&0 \\ &&\omega_2 \end{matrix} \end{equation*} \notag $$
The class $x^2_{+}$ represents the 5-fold Massey product $\langle \omega_1,\omega_2,-2\omega_1,\omega_1,\omega_2\rangle$ with the defining system
$$ \begin{equation*} \begin{matrix} \omega_1 & \omega_3&\hphantom{-}2\omega_4&-\omega_5& \\ &\omega_2&\hphantom{-} 2\omega_3&-\omega_4&0 \\ &&-2\omega_1&\hphantom{-}\omega_2&-\omega_4 \\ &&&\hphantom{-}\omega_1&\hphantom{-}\omega_3 \\ &&&&\hphantom{-}\omega_2 \end{matrix} \end{equation*} \notag $$

Now note that the class $x^2_-$ in the trigraded Buchstaber spectral sequence is represented by $d_2\bigl(-(e_1e_2)^*\otimes \omega_2\bigr)$. Indeed, the differential $d_1$ takes $-(e_1e_2)^*\otimes \omega_2\in E_1^{2,*,*}$ to $-(e_2)^*\otimes \omega_1\wedge \omega_2$, which is zero in $E_1$ since it is the image of the differential of $(e_2)^*\otimes \omega_3$. Therefore, the cochains

$$ \begin{equation*} -(e_1e_2)^*\otimes \omega_2\quad\text{and}\quad -(e_1e_2)^*\otimes \omega_2- (e_2)^*\otimes \omega_3 \end{equation*} \notag $$
represent the same class in $E_1^{*,*,*}$. As we just saw,
$$ \begin{equation*} d_{\rm CE}\bigl(-(e_1e_2)^*\otimes \omega_2- (e_2)^*\otimes \omega_3\bigr)=\omega_3\wedge\omega_2, \end{equation*} \notag $$
which implies that
$$ \begin{equation*} x^2_-= d_2 (-(e_1e_2)^*\otimes \omega_2). \end{equation*} \notag $$
It is similarly verified that the class $[(e_5)^*\otimes \omega_2+ 6(e_3^2)^*\otimes \omega_1]\in E^{4,*,*}_1$ lies in the kernel of the differentials $d_2$ and $d_3$, and the differential $d_4$ takes it to $x^2_+$, which is represented by the cocycle $\omega_5\wedge\omega_2-3\omega_4\wedge\omega_3$. To do this, observe that the cochains
$$ \begin{equation*} (e_5)^*\otimes \omega_2+ 6(e_3^2)^*\otimes \omega_1 \quad \text{and} \quad (e_5)^*\otimes \omega_2+6(e_3^2)^*\otimes \omega_1-3 e_4^* \otimes \omega_3 \end{equation*} \notag $$
represent the same class in $E^{4,*,*}_1$, and that
$$ \begin{equation*} d_{\rm CE}\bigl((e_5)^*\otimes \omega_2+ 6(e_3^2)^*\otimes \omega_1- 3 e_4^* \otimes \omega_3\bigr)=\omega_5\wedge\omega_2-3\omega_4\wedge\omega_3. \end{equation*} \notag $$

6. Massey products and the Eilenberg–Moore spectral sequence ($\operatorname{EMss}$)

6.1. Scalar and matric Massey products

For the convenience of the reader and to fix the notation, we include here the definitions of $n$-fold Massey products, both scalar and matric, and also give their properties that we need. For the relation to formal connections and the Maurer–Cartan equation, see [14]. Further details can be found in [68] and [13]. We choose the signs following [14]: $\overline{x}=(-1)^k x$ if $x$ has degree $k$.

Let $C^*$ be a differential graded algebra over a ring $R$, where the differential $d$ is assumed to raise the grading by one. In this section we denote the cohomology of $C^*$ with respect to $d$ by $H^*(C^*)=H^*(C^*,d)$, this will not interfere with the notation for $\operatorname{Ext}^{*,*}_{C^*}(R,R)$.

6.1.1. Scalar triple Massey product

Let $a_1,a_2,a_3\in H^*(C^*)$ be homogeneous classes satisfying $\overline{a}_1a_2=0$ and $\overline{a}_2a_3=0$. Let $x_i\in C^*$ be a cocycle representing $a_i\in H^*(C^*)$, for $i=1,2,3$. Then $\overline{x}_1x_2$ and $\overline{x}_2x_3$ are cobounaries, so there exist elements $U$ and $V$ such that

$$ \begin{equation*} dU=\overline{x}_1x_2\quad\text{and}\quad dV=\overline{x}_2x_3. \end{equation*} \notag $$
It is easy to check that $\overline{U}x_3+\overline{x}_1 V$ is a cocycle whose cohomology class depends in general on the choice of representatives $x_1$, $x_2$, $x_3$ and elements $U$ and $V$. The triple Massey product is the set of cohomology classes
$$ \begin{equation*} \langle a_1,a_2,a_3\rangle=\{[\overline{U}x_3+\overline{x}_1 V]\,{\in}\,H^*(C^*) \colon[x_i]=a_i\,{\in}\,H^*(C^*),dU=\overline{x}_1x_2,dV=\overline{x}_2x_3\}. \end{equation*} \notag $$
As follows from the definition, this operation is partially defined and multivalued. Indeed, the cohomology class of $[a]$ depends on the choice of the elements $U$ and $V$, and for a different choice of these elements the class $[a']$ defined by them differs from $[a]$. More precisely,
$$ \begin{equation*} [a]-[a']\in H^{|a_1|+|a_2|-1}(C^*) a_3+a_1 H^{|a_2|+|a_3|-1}(C^*), \end{equation*} \notag $$
and any element of this $R$-module can be realised in this way.

6.1.2. The scalar $n$-fold Massey product

Consider the cohomology classes $a_i\in H^{p_i}(C^*)$. A set of elements $x(i,j)\in C^*$, where $1\leqslant i\leqslant j\leqslant n$ and $(i,j)\ne(1,n)$, is called a defining system for the classes $a_i$, $i=1,\dots,n$, if the following conditions are satisfied:

Then $x(1,n)=\displaystyle\sum_{k=1}^{n-1}\overline{x(1,k)}\,x(k+1,n)$ is a cocycle, and the set of cohomology classes represented by the elements $x(1,n)$ corresponding to all defining systems $\{x(i,j)\}$ for $a_1,\dots,a_n$, is called the $n$-fold Massey product. It is also partially defined and multivalued. For fixed $a_1,\dots,a_n$, the differences of two elements of the form $[x(1,n)]$ corresponding to all possible pairs of defining systems form a group, just as for $n=3$, although its description is more complicated. Furthermore, already in the description of the indeterminacy of a 4-fold Massey scalar product, matric products naturally arise; see the next paragraph.

An $n$-fold Massey product is trivial if it contains zero.

Remark 3. It is convenient to consider the two-fold Massey product. It is defined uniquely for all classes $a,b\in H^*(C^*)$ and coincides up to a sign with the ordinary cohomology product: $\langle a,b \rangle=\overline{a}\, b$.

6.1.3. Matric $n$-fold Massey product

There is a generalisation of $n$-fold Massey products in which $x(i,j)$ are matrices of homogeneous elements of the augmentation ideal of $C^*$. Condition (M3) from the previous paragraph imposes two types of consistency conditions on the gradings of the matrix entries $x(i,j)$. Namely, let

$$ \begin{equation*} A=(a_{ij}\in C^*\colon 1\leqslant i \leqslant m, 1\leqslant j\leqslant l), \end{equation*} \notag $$
and
$$ \begin{equation*} B=(b_{jk}\in C^*\colon 1\leqslant j \leqslant l', 1\leqslant k\leqslant n). \end{equation*} \notag $$
We say that $A$ and $B$ are multiplicable if

(1) $l=l'$;

(2) $|a_{ij}|+|b_{jk}|$ does not depend on $j$, for any $1\leqslant i \leqslant m$ and $1\leqslant k\leqslant n$.

For multiplicable matrices $A$ and $B$ the product $AB$ is defined and its elements are homogeneous elements of $C^*$.

With each matrix $A=(a_{ij}\in C^*\colon 1\leqslant i \leqslant m, 1\leqslant j\leqslant l)$ the integer matrix $D(A)=(|a_{ij}|\colon 1\leqslant i \leqslant m, 1\leqslant j\leqslant l)$ is associated. For multiplicable marices $A$ and $B$ we have

$$ \begin{equation*} D(AB)=D(A)*D(B)=(|a_{ij}|+|b_{jk}|\colon 1\leqslant i \leqslant m, 1\leqslant k\leqslant n). \end{equation*} \notag $$

Let $A_1,\dots,A_n$ be a set of matrices consisting of homogeneous elements of $H^*(C^*)$, where $A_i$ and $A_{i+1}$ are multiplicable for all $i=1,\dots,n$. We say that a set of matrices $X(i,j)$, $1\leqslant i\leqslant j \leqslant n$, $(i,j)\ne (1,n)$, consisting of homogeneous elements of $C^*$, forms a defining system if

(M1) for all $i$ the matrix $X(i,i)$ consists of cocycles representing the corresponding elements of the matrix $A_i$;

(M2) the gradings of the entries of the matrix $X(i,j)$ are related to the gradings of the entries of $A_i,A_{i+1},\dots,A_j$ as follows: the entries of the matrix $D(X(i,j))$ are less than the corresponding entries of the matrix $D(A_i)*D(A_{i+1}) *\cdots * D(A_j)$ by exactly $j-i+1$;

(M3) $d X(i,j)=\displaystyle\sum_{k=i}^{j-1}\overline{X(i,j)}\, X(k+1,j)$.

If the matrices $X(i,j)$ form a defining system, then the matrix

$$ \begin{equation*} C(\{A_i\},\{X(i,j)\})=\sum_{k=1}^{n-1}\overline{X(i,j)}\, X(k+1,j) \end{equation*} \notag $$
consists of cocycles. The set of matrices consisting of cohomology classes represented by all possible $C(\{A_i\},\{X(i,j)\})$ for fixed $A_1,\dots,A_n$ is called the $n$-fold matric Massey product and is denoted by $\langle A_1,A_2,\dots,A_n\rangle$. In the case where $A_1$ is a row and $A_n$ is a column, the Massey product $\langle A_1,A_2,\dots,A_n\rangle$ consists of matrices of size $1\times 1$, that is, of cohomology classes.

It is convenient to write the defining system $X(i,j)$, $i\leqslant j$, $(i,j)\ne (1,n)$, as a matrix of size $n\times n$ which has no elements in the upper right corner or under the diagonal, and $X(i,j)$ is at the position $(i,j)$.

Remark 4. The conditions (M3) for both scalar and matric defining systems can be written in the form of the Maurer–Cartan equation; this concept was developed in [14] and found important applications. Consider the following matrix:

$$ \begin{equation*} \mathbb{X}=\begin{pmatrix} 0 & X(1,1) & X(1,2) & \dots & X(1,n) \\ & 0 & X(2,2) & \dots & X(2,n) \\ & & \ddots & \ddots & \vdots \\ & {\large\textbf{0}} & & \ddots & X(n,n) \\ & & & & 0 \end{pmatrix}. \end{equation*} \notag $$
Then the conditions (M3) are equivalent to the Maurer-Cartan equation
$$ \begin{equation*} d\mathbb{X}-\overline{\mathbb{X}}\cdot\mathbb{X}=0. \end{equation*} \notag $$

Remark 5. The two-fold matric Massey product is defined for any multiplicable row $(a_1,\dots, a_n)$ and column $(b_1,\dots, b_n)^\top$:

$$ \begin{equation*} \langle(a_1,\dots,a_n),(b_1,\dots,b_n)^\top\rangle= \sum_{k=1}^n \overline{a}_k\, b_k. \end{equation*} \notag $$

The indeterminacy of the matric Massey product $\langle A_1,A_2,\dots,A_n\rangle$ is the group

$$ \begin{equation*} \operatorname{In}\langle A_1,A_2,\dots,A_n\rangle= \{x-y\colon x,y\in \langle A_1,A_2,\dots,A_n\rangle\}. \end{equation*} \notag $$
A result due to Kraines describes the indeterminacy of the $n$-fold matric Massey product as a subset in the union of some $(n-1)$-fold matric Massey products (see Proposition 2.3 in May’s paper [13]). We do not need a more precise formulation, so we do not present it. We note, however, that even for scalar $n$-fold products, where $n\geqslant 4$, their indeterminacy is necessarily expressed in terms of $(n-1)$-fold matric Massey products.

6.2. Algebraic construction of $\operatorname{EMss}$

Consider a differential graded algebra ${\mathcal A}$ over a field $\Bbbk$ with differential $d_{\mathcal A}$ and augmentation $\varepsilon\colon {\mathcal A}\to\Bbbk$. We assume that the grading is ${\mathcal A}=\bigoplus\limits_{j=0}^\infty{\mathcal A}^j$, where $d_{\mathcal A}$ increases the grading by one, and the augmentation $\varepsilon\colon {\mathcal A}\to \Bbbk$ defines an isomorphism $\varepsilon \colon {\mathcal A}^0\to \Bbbk$.

Let $M$ be a right differential graded module over ${\mathcal A}$, and let $N$ be a left differential graded module over ${\mathcal A}$, both concentrated in non-negative degrees.

In this situation the functor $\operatorname{Tor}_{\mathcal A}^{*}(M,N)$ is defined. A detailed exposition within the general theory of derived functors can be found, for example, in [5]. The functor $\operatorname{Tor}$ is defined for differential graded algebras and modules in a more general situation, but we restrict ourselves to the case described above.

There is a convenient description of $\operatorname{Tor}_{\mathcal A}^{*}(M,N)$ in terms of the bar resolution. Let

$$ \begin{equation} B^{-n,t}=B^{-n,t}(M,{\mathcal A},N)= (M\otimes {\mathcal A}^{\otimes n} \otimes N)^t. \end{equation} \tag{30} $$
Two differentials are defined on the groups $B^{-n,t}$: the internal differential $d_I$ and the combinatorial differential $d_B$, given by the formulae
$$ \begin{equation} \begin{aligned} \, d_B( x[a_1,\dots, a_n ] y) &= (-1)^{|x|} (xa_1) [a_2,\dots, a_n] \\ &\qquad+\sum_{k=1}^{n-1} (-1)^{\varepsilon_B} x[a_1,\dots, a_ka_{k+1},\dots, a_n] y \\ &\qquad+(-1)^{|x|+\sum\limits_{j=1}^n|a_j|+n} x[a_1,\dots, a_{n-1}] a_{n} y, \\ d_I( x[a_1,\dots, a_n ] y) &= (d_M(x)) [a_1,\dots, a_n] + \sum_{k=1}^{n} (-1)^{\varepsilon_I} x[a_1,\dots, d a_k,\dots, a_n] y \\ &\qquad+ (-1)^{|x|+\sum\limits_{j=1}^n|a_j|+n} x[a_1,\dots, a_{n}] d_N y, \end{aligned} \end{equation} \tag{31} $$
where
$$ \begin{equation} \varepsilon_B=|x|+\sum_{j=1}^{k}|a_j|+k\quad\text{and}\quad \varepsilon_I=|x|+\sum_{j=1}^{k-1}|a_j|+k-1. \end{equation} \tag{32} $$

The differentials $d_I$ and $d_B$ anticommute, so we have a complex

$$ \begin{equation*} (\operatorname{Tot} B(M,{\mathcal A},N))^n= \bigoplus_{i+j= n}B^{i,j}(M,{\mathcal A},N) \end{equation*} \notag $$
with differential $D=d_I+d_B$. Its cohomology is by definition equal to $\operatorname{Tor}^{*}_{\mathcal A}(M,N)$.

The Eilenberg–Moore spectral sequence $\operatorname{EMss}$ is defined by one of the standard filtrations of the bicomplex $B^{i,j}(M,{\mathcal A},N)$:

$$ \begin{equation*} F^{-p}(\operatorname{Tot} B(M,{\mathcal A},N))^n= \bigoplus_{i+j=n,\, i\geqslant -p}B^{i,j}(M,{\mathcal A},N). \end{equation*} \notag $$
It is easy to check that the first term of $\operatorname{EMss}$ is isomorphic to
$$ \begin{equation*} E_1^{*,*}=H(B^{*,*}(M,{\mathcal A},N),d_I). \end{equation*} \notag $$
Since the modules are over the field $\Bbbk$, the first term $E_1^{*,*}$ is isomorphic to the bar construction of the algebra $H({\mathcal A})$ with coefficients in the modules $H(M)$ and $H(N)$:
$$ \begin{equation*} E_1^{-p,q}=B^{-p,q} (H(M), H({\mathcal A}), H(N)). \end{equation*} \notag $$
Finally, since the differential in $E_1^{p,q}$ is induced by the differential $d_B$, we have
$$ \begin{equation*} E_2^{-p,q}=\operatorname{Tor}^{-p,q}_{H({\mathcal A})}(H(M),H(N)). \end{equation*} \notag $$

Other aspects of the algebraic construction of $\operatorname{EMss}$ can be found in a number of works including [15] and [16].

6.3. Differentials in $\operatorname{EMss}$ and Massey products

In this subsection we discuss how matric Massey products manifest themselves in the Eilenberg–Moore spectral sequence and why elements of the groups $\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$, $s\geqslant 2$, decompose into iterated Massey products of elements of $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$.

A number of authors have developed the idea of a connection between differentials of the Eilenberg–Moore spectral sequence and Massey products. We do not aim to describe the history of this issue in detail, so we only mention the monograph [17] and the references there.

For simplicity we denote the cohomology with respect to the inner differential by $H{\mathcal A}$ instead of $H^*({\mathcal A},d_{\mathcal A})$. The same applies to modules $M $ and $N$.

Consider the homomorphism $\gamma\colon \operatorname{Tor}^*_A(M,N)\to H(M\otimes_{\mathcal A} N)$ defined at the level of complexes by the map

$$ \begin{equation*} B(M,{\mathcal A},N)= M\otimes_{\mathcal A} B({\mathcal A},{\mathcal A},N)\to M\otimes_{\mathcal A} N. \end{equation*} \notag $$
It is easy to see that if the inner differentials in ${\mathcal A}$, $M$, and $N$ are trivial, then $\gamma$ defines the standard isomorphism $\operatorname{Tor}_{\mathcal A}^0(M,N)\to M\otimes_{\mathcal A}N$.

Theorem 11 ([17], § 5.9). The image of $\gamma$ consists of elements that are represented by all possible matric Massey products $\langle A_0,A_1,\dots,A_{p+1}\rangle$, $p\geqslant 0$, where $A_0$ is a row vector of elements of $HM$, $A_1,\dots,A_p$ are matrices of elements of $H{\mathcal A}$, and $A_{p+1}$ is a column vector of elements of $HN$.

The algebra ${\mathcal A}$ is equipped with an augmentation, so we can regard $\overline{{\mathcal A}}=\ker\varepsilon \colon {\mathcal A}\to\Bbbk$ as a module $M$ (or $N$). Let $\overline{HA}=\ker \varepsilon^*\colon HA\to \Bbbk$. Then the inclusion $i\colon \overline{{\mathcal A}}\to {\mathcal A}$ induces a homomorphism

$$ \begin{equation*} i^*=H(i\otimes\operatorname{id})\colon H(\bar{A}\otimes_{\mathcal A}N) \to H(A\otimes_{\mathcal A}N)=H(N). \end{equation*} \notag $$
We denote the image of the composite
$$ \begin{equation*} i^*\circ \gamma\colon \operatorname{Tor}^*_{\mathcal A}(\overline{{\mathcal A}},N)\to H(\overline{{\mathcal A}}\otimes_{\mathcal A} N)\to HN \end{equation*} \notag $$
by $D(H{\mathcal A},HN)$.

Consider the edge homomorphism

$$ \begin{equation*} \pi\colon HN\to \operatorname{Tor}^*_{\mathcal A}(\Bbbk,N) \end{equation*} \notag $$
defined as the composite
$$ \begin{equation*} HN=\Bbbk\otimes HN\to \Bbbk\otimes_{H{\mathcal A}} HN=E_2^{0,*}\to E^{0,*}_\infty \to \operatorname{Tor}^*_{{\mathcal A}}(\Bbbk,N). \end{equation*} \notag $$

Corollary 3 ([17], § 5.12). The kernel of $\pi\colon HN\to \operatorname{Tor}^*_{{\mathcal A}}(M,N)$ coincides with $D(H{\mathcal A},HN)$.

Next, we have the suspension homomorphism (see [17], § 3.7)

$$ \begin{equation*} \sigma\colon \overline{H{\mathcal A}}\to E^{-1,*}_2\to E^{-1,*}_\infty \subset \operatorname{Tor}_{{\mathcal A}}(\Bbbk,\Bbbk). \end{equation*} \notag $$
Consider another inclusion $i_1\colon\overline{\mathcal A}\otimes_{\mathcal A} \overline{{\mathcal A}} \to {\mathcal A}\otimes_{\mathcal A}{\mathcal A}={\mathcal A}$. It induces a homomorphism $i_1^*\colon H(\overline{\mathcal A}\otimes_{\mathcal A} \overline{{\mathcal A}}) \to H({\mathcal A})$. We denote the image of the composite
$$ \begin{equation*} i_1^*\circ\gamma\colon \operatorname{Tor}^*_{\mathcal A}(\overline{{\mathcal A}}, \overline{{\mathcal A}})\to H(\overline{\mathcal A} \otimes_{\mathcal A}\overline{{\mathcal A}}) \to H({\mathcal A}) \end{equation*} \notag $$
by $DH({\mathcal A})$.

Corollary 4 ([17], § 5.13). The kernel of $\sigma\colon \overline{H\!{\mathcal A}}\to \operatorname{Tor}_{{\mathcal A}}(\Bbbk,\Bbbk)$ coincides with $DH({\mathcal A})$.

These results can be used to prove that each element of $\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$, $s\geqslant 2$, can be decomposed into an iterated Massey product of elements of $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$, where $A$ is a connected graded algebra with augmentation $\varepsilon\colon A\to \Bbbk$. We take the cobar construction $F(A_*)$ of its dual coalgebra as the differential graded algebra ${\mathcal A}$. Then

$$ \begin{equation*} H(F(A_*))=\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk). \end{equation*} \notag $$
For simplicity we write $H^*(A)$ instead of $H(F(A_*))$ for the rest of this section.

Consider the $\operatorname{EMss}$ converging to $\operatorname{Tor}^*_{F(A_*)}(F(A_*),\Bbbk)$ with the second term $\operatorname{Tor}^*_{H^*(A)}(H^*(A),\Bbbk)$.

It can be verified that the group $\operatorname{Tor}^0_{F(A_*)}(F(A_*),\Bbbk)$ is isomorphic to $\Bbbk$ and

$$ \begin{equation*} \operatorname{Tor}^p_{F(A_*)}(F(A_*),\Bbbk)=0 \end{equation*} \notag $$
for $p>0$. This would be obvious if the differential in $F(A_*)$ were trivial.

Let $QH^*A=\overline{H^*A}/(\overline{H^*A})^2$ be the module of indecomposables. There is a commutative diagram

Recall that $\operatorname{Tor}^A_i(\Bbbk,\Bbbk)=QA= \bar{A}/(\bar{A})^2$. Furthermore, the edge homomorphism $e$ is an isomorphism on $E_2^{-1,1}$ and on $E_2^{0,0}$. Hence
$$ \begin{equation*} H^1A=(QA)^*=\overline{H^*A}/\ker\sigma, \end{equation*} \notag $$
which is precisely the set of elements that are not decomposable into matric Massey products; the details can be found in [17].

7. Nilmanifolds $M^n$

7.1. Homogeneous manifolds of the group of polynomial transformations of the line

In this and the next sections we apply the $\operatorname{Bss}$ to the description of the cohomology ring of nilmanifolds.

Recall that a nilmanifold is a compact homogeneous space of a real finite-dimensional simply connected nilpotent Lie group. Malcev [69] showed that, given a simply connected nilpotent Lie group, a nilmanifold exists if and only if the corresponding Lie algebra has a basis with rational structure constants.

In [40] the following structure of a nilpotent Lie group on $\mathbb{R}^n$ was introduced. Let

$$ \begin{equation*} L^n=\biggl\{p_x(t)=t+\sum_{k=1}^n x_k t^{k+1}, \ x_k\in\mathbb{R}\biggr\}. \end{equation*} \notag $$
As a manifold, the space $L^n$ is diffeomorphic to $\mathbb{R}^n$. The product of two elements of $L^n$ is defined by composition:
$$ \begin{equation*} (p_x * p_y)(t)=p_z(t)=p_y(p_x(t))\mod t^{n+2}. \end{equation*} \notag $$

There is a lattice $\Gamma^n$ in $L^n$ consisting of elements $p_x(t)$ with integer coefficients $x_k$. We obtain a smooth closed nilmanifold $M^n= L^n/\Gamma^n$, where $\Gamma^n$ acts on $L^n$ by right shifts.

The group homomorphism $L^k\to L^{k-1}$ omitting the last coordinate $x_k$ generates two towers of groups

$$ \begin{equation*} \cdots \to L^n\to L^{n-1}\to\cdots \to L^1 \quad\text{and}\quad \cdots \to \Gamma^n\to \Gamma^{n-1}\to\cdots \to \Gamma^1 \end{equation*} \notag $$
and a tower of bundles
$$ \begin{equation} \cdots \to M^n\to M^{n-1}\to \cdots \to M^1=S^1 \end{equation} \tag{33} $$
with fibre $S^1$. Note that there are other lattices in the groups $L^n$ that are compatible with projections and thus generate a tower of bundles of nilmanifolds. By Nomizu’s theorem the real cohomology of a nilmanifold can be computed by using invariant differential forms on the corresponding nilpotent Lie group and therefore does not depend on the particular choice of a lattice. Other interesting lattices in the groups $L^n$ were used in [39] and [14].

It is easy to see that $M^2$ is a two-dimensional torus. The three-dimensional manifold $M^3$, the Heisenberg manifold, is not diffeomorphic to a torus.

In the Lie algebra $\mathcal{G}(L^n)$ there exists a basis $e_1,\dots,e_n$ with integer structure constants, namely,

$$ \begin{equation*} [e_i,e_j]=(j-i) e_{i+j}\quad\text{for}\ \ i+j\leqslant n,\quad\text{and}\quad [e_i,e_j]=0\quad\text{for}\ \ i+j >n. \end{equation*} \notag $$
Clearly, $\mathcal{G}(L^n)$ is isomorphic to the quotient of the Lie algebra $L_1$ considered above by the ideal generated by the $e_k$ with $k\geqslant n+1$. This implies a connection between the series of nilmanifolds $M^n$ and the Landweber–Novikov algebra.

There are other important properties of the nilmanifolds $M^n$, which are beyond the scope of this article; we refer to [60] and [40]. For the connections with the Landweber–Novikov algebra, as well as with the group of diffeomorphisms of the line, see [34] and the survey article [70].

According to Nomizu’s theorem [41], the de Rham cohomology algebra of a nilmanifold is isomorphic to the cohomology of the corresponding nilpotent Lie algebra. A question arises naturally: what information about the cohomology ring of a nilmanifold is carried by the $\operatorname{Bss}$ of the corresponding Lie algebra?

We present a solution for nilmanifolds of dimension $3$ and $4$. As we will see, the problem is non-trivial even in these cases. We give a fairly detailed solution in these dimensions to demonstrate the general approach in the case of higher dimension.

To complete the introduction to this section, we observe the following. Consider the exterior algebra $\Lambda(\mathcal{G}L^n)^*$ generated by the elements $\omega_1,\dots,\omega_n$ dual to $e_1,\dots,e_n$. By Nomizu’s theorem the cohomology of $M^n$ is isomorphic to the cohomology of the complex $\Lambda (\mathcal{G}L^n)^*$ with respect to the differential $d_{\rm CE}$ given on the generators by

$$ \begin{equation*} d_{\rm CE}\,\omega_k=\sum_{i<k/2}(k-2i)\,\omega_{k-i}\wedge\omega_i. \end{equation*} \notag $$
If we assign the bigrading $(1,-2k)$ to $\omega_k$, then the differential $d_{\rm CE}$ becomes homogeneous with respect to the second grading. Hence the second grading descends to the cohomology, which gives a natural bigrading on the cohomology of the manifolds $M^n$; details can be found in [60]. This property is important since it implies that the $\operatorname{Bss}$ is trigraded, in contrast to the general case of nilpotent Lie algebras (see § 3.7).

The existence of a bigrading is due to the fact that if we assign a second grading $2k$ to the $k$th basis element $e_k$ of the Lie algebra $\mathcal{G}L^n$, then the commutator becomes homogeneous with respect to it. We choose the additional grading to be $2k$ instead of $k$ because then it does not affect the sign in formulae like $\theta\wedge\eta=\pm\eta\wedge\theta$.

Using the classification of low-dimensional nilpotent real Lie algebras, one can directly verify that that the homogeneity of the commutator holds for all algebras of dimension at most $5$ (in some cases, it is necessary to reorder the basis elements). On the other hand, in dimension $6$ there is a nilpotent Lie algebra generated by the elements $e_k$, $1\leqslant k\leqslant 6$, with the relations $[e_1,e_2]=e_3$, $[e_1,e_3]=e_6$, and $[e_4,e_5]=e_6$ for which the commutator cannot be made homogeneous by renumbering the basis elements.

We denote the differential $d_{\rm CE}\colon\Lambda\mathfrak{g}^*\to\Lambda\mathfrak{g}^*$ simply by $d$. The necessary information about Massey products is presented in § 9.

7.2. The Lie algebra $\mathcal{G}L^3$

The Lie algebra $\mathcal{G}L^3$ coincides with the well-known Lie–Heisenberg algebra. It has three generators $e_1$, $e_2$, and $e_3$ and the commutation relations

$$ \begin{equation*} [e_1,e_2]=e_3 \quad\text{and}\quad [e_2,e_3]=[e_1,e_3]=0. \end{equation*} \notag $$
The cohomology $H^*(\mathcal{G}L^3)$ is easy to obtain from the Chevalley–Eilenberg complex. Here are $\mathbb{R}$-bases for the cohomology $H^k(\mathcal{G}L^3)$ (we denote by $\omega_k$ the element dual to $e_k$):
$$ \begin{equation*} \begin{alignedat}{2} &k=0: &&\quad 1; \\ &k=1: &&\quad \omega_1,\ \ \omega_2; \\ &k=2: &&\quad \omega_3\wedge \omega_1,\ \ \omega_3\wedge \omega_2; \\ &k=3: &&\quad \omega_3\wedge \omega_2\wedge \omega_1. \end{alignedat} \end{equation*} \notag $$
It is easy to see that, except for those products of elements of complementary dimensions that are non-trivial by Poincaré duality, all other products in $H^*(\mathcal{G}L^3)$ are trivial.

On the other hand, the $2$-dimensional classes are realised by non-trivial triple Massey products. Indeed

$$ \begin{equation*} \omega_3\wedge \omega_1 = \langle \omega_1,\omega_1,\omega_2\rangle, \end{equation*} \notag $$
and
$$ \begin{equation*} \omega_3\wedge\omega_2 =\langle - \omega_1,\omega_2,\omega_2\rangle. \end{equation*} \notag $$

Recall the notation $\overline x=(-1)^q x$ for an element $x$ of grading $q$. To check the first equality, set $u=0$, $v=\omega_3$. Then

$$ \begin{equation*} du=0=\overline{\omega}_1\wedge\omega_1\quad\text{and}\quad dv=\omega_2\wedge\omega_1=\overline{\omega}_1\wedge\omega_2. \end{equation*} \notag $$
This implies that $\overline{u}\wedge \omega_2 + \overline{\omega}_1\wedge v= \omega_3\wedge\omega_1$, proving the first equality.

To check the second equality, set $u=-\omega_3$ and $v=0$. Then

$$ \begin{equation*} dv=0=\overline{\omega}_2\wedge\omega_2\quad\text{and}\quad du=-\omega_2\wedge\omega_1=(-\overline{\omega}_1)\wedge\omega_2. \end{equation*} \notag $$
This implies that $\overline{u}\wedge\omega_2+\overline{\omega}_1\wedge v= \omega_3\wedge\omega_2$, as required.

Now we describe the $\operatorname{Bss}$ for the Hopf algebra $U\mathcal{G}L^3$. Further on in this section we use the notation $\mathfrak{g}=\mathcal{G}L^3$.

The basis of the universal enveloping algebra $U\mathfrak{g}$ consists of monomials $e_1^a e_2^b e_3^c$. We put

$$ \begin{equation*} p(e_1^a e_2^b e_3^c)=a+b+2c. \end{equation*} \notag $$

Lemma 8. (a) An element $x=e_1^a e_2^b e_3^c$ such that $p(x)=p$ belongs to $N_p$ and does not belong to $N_{p-1}$.

(b) Furthermore, $N_{p }=N_{p-1} \oplus \mathbb{Q}\langle e_1^a e_2^b e_3^c\colon a+b+2c=p\rangle$.

Proof. This is similar to the proof of Theorem 7. $\Box$

Corollary 5. The generating series of the sequence $k_p=\dim N_p/N_{p-1}$ is $1/((1-x)^2(1-x^2))$. In particular,

$$ \begin{equation*} k_{2n }=\dim N_{2n}/N_{2n-1}=(n+1)^2\quad\textit{and}\quad k_{2n+1}=\dim N_{2n+1}/N_{2n}=(n+1)(n+2). \end{equation*} \notag $$

Now we consider the complex $(U\mathfrak{g})^*\otimes\Lambda\mathfrak{g}^*$ dual to $U\mathfrak{g} \otimes \Lambda \mathfrak{g}$. We denote by $\omega_i$ the element of $\mathfrak{g}^*$ dual to $e_i$. Then $d\omega_k$ is given by the formula (29), in particular,

$$ \begin{equation*} d\omega_1=d\omega_2=0\quad\text{and}\quad d\omega_3=\omega_2\wedge\omega_1. \end{equation*} \notag $$

It is easy to describe the differential $d_{\rm CE}$ of the complex $(U\mathfrak{g})^* \otimes \Lambda \mathfrak{g}^*$ at any monomial $(e_1^a e_2^be_3^c)^* \otimes \omega$ (here $\omega \in \Lambda \mathfrak{g}^*$):

$$ \begin{equation} \begin{aligned} \, \notag d_{\rm CE}((e_1^a e_2^be_3^c)^* \otimes \omega)&= (e_1^{a-1} e_2^be_3^c)^* \otimes \omega_1\wedge \omega + (e_1^a e_2^{b-1}e_3^c)^* \otimes\omega_2\wedge \omega \\ \notag &\qquad+(e_1^a e_2^be_3^{c-1})^* \otimes\omega_3\wedge \omega - (b+1) (e_1^a e_2^{b+1}e_3^{c-1})^* \otimes \omega_1\wedge\omega \\ &\qquad+(e_1^a e_2^be_3^c)^* \otimes d\omega. \end{aligned} \end{equation} \tag{34} $$
If some of the exponents $a$, $b$, and $c$ are zero, then the corresponding terms on the right-hand side are omitted.

We collect together the properties of the differentials in the $\operatorname{Bss}$ for the Lie algebra $\mathfrak{g}=\mathcal{G}L^3$:

Statement 2. (a) The differential $d_1\colon N_p/N_{p-1}\to N_{p-1}/N_{p-2}\otimes H^1(\mathfrak{g})$ is injective. For $p=1$ it is an isomorphism.

(b) The differential $d_1\colon N_{p-1}/N_{p-2}\otimes H^1(\mathfrak{g})\to N_{p-2}/N_{p-3}\otimes H^2(\mathfrak{g})$ is zero.

(c) The differential $d_1\colon N_{p-2}/N_{p-3}\otimes H^2(\mathfrak{g})\to N_{p-3}/N_{p-4}\otimes H^3(\mathfrak{g})$ is surjective for $p\geqslant 3$.

Proof. Parts (a) and (b) follow from the general properties of the differential $d_1$ (see Theorem 2).

To prove part (c) note that $(e_1^a e_2^be_3^c)^* \otimes\omega_3\wedge\omega_2\wedge\omega_1$ is equal to $d_1((e_1^a e_2^{b+1}e_3^c)^* \otimes\omega_1\wedge\omega_3)$. $\Box$

Statement 2 gives us sufficient information for the calculation of both $E_2^{*,*,*}$ and $d_2$. In fact, the only non-trivial terms in $E_2^{*,*,*}$ are $E^{q-2,-q,*}_2$, $q\geqslant 2$, and $E^{p,-p-1,*}_2$, $p\geqslant 2$. Statement 2 implies that their dimensions are given by

$$ \begin{equation*} \begin{aligned} \, \dim E^{q-2,-2,*}_2&=\dim E^{q-2,-q}_1-\dim E^{q-3,-q}_1=2 k_{q-2}-k_{q-3}, \\ \dim E^{p,-p-1,*}_2&=\dim E^{p,-p-1,*}_1-\dim E^{p+1,-p-1,*}_1= 2k_{p}-k_{p+1}. \end{aligned} \end{equation*} \notag $$

Statement 3. The equality $\dim E^{p,-p-1,*}_2=\dim E^{p-2,-p,*}_2$ holds for $p\geqslant 2$.

Proof. We need to prove that $2k_{p}-k_{p+1}=2 k_{q-2} - k_{q-3}$ for $p\geqslant 2$. Let
$$ \begin{equation*} f(x)=\dfrac{1}{(1-x)^2(1-x^2)} \end{equation*} \notag $$
be the generating series of the sequence $(k_p)$. Then we need to prove that the coefficients of $x^k$ in $(1-2x+2x^3-x^4) f(x)$ is zero for $k\geqslant 3$. It is easy to verify that, in fact, $(1-2x+2x^3-x^4) f(x)=1$. $\Box$

Theorem 12. In $\operatorname{Bss}$ for the Lie algebra $U\mathcal{G}L^3$, the differential $d_2$ is an isomorphism, except on the group $E_2^{0,0,0}$. In particular, $E_3^{*,*,*}= E_\infty^{*,*,*}$.

Proof. The only possibly non-trivial differentials are $d_2\colon E^{p,-p-1,*}_2\to E^{p-2,-p,*}_2$, $p\geqslant 2$. We prove that all of them are isomorphisms using induction for all $p\geqslant 2$. For $p=2$ the differential $d_2\colon E^{p,-p-1,*}_2\to E^{p-2,-p,*}_2$ is injective by dimensional considerations. The dimensions of these two spaces coincide by Statement 3. Therefore, $d_2$ is an isomorphism for $p=2$. Then we go over to $p=3$, and so on. $\Box$

Remark 6. (a) Theorem 12 can also be derived from Theorem 5.

(b) In dimension 3, there are only two nilpotent Lie algebras: the commutative Lie algebra and $\mathcal{G}L^3$. The first case was considered in § 4.

7.3. The Lie algebra $\mathcal{G}L^4$

We begin this paragraph with the following observation. The classification of four-dimensional nilpotent real Lie algebras is simple. There are only three such algebras:

$\bullet$ commutative;

$\bullet$ a direct sum of a one-dimensional algebra and $\mathcal{G}L^3$;

$\bullet$ the algebra $\mathcal{G}L^4$.

The case of commutative Lie algebra was considered in § 4. The algebra $\mathbb{R}\mathbin{\oplus}\mathcal{G}L^3$ corresponds to the well-known Kodaira–Thurston manifold. The universal enveloping algebra of $\mathbb{R}\mathbin{\oplus}\mathcal{G}L^3$ is isomorphic to the tensor product of the universal enveloping algebras of the terms, and the Buchstaber filtration on $U(\mathbb{R}\mathbin{\oplus}\mathcal{G}L^3)^*$ is induced by the corresponding filtrations on $U(\mathbb{R})^*$ and $U(\mathcal{G}L^3)^*$. This allows one to reduce the description of the trigraded spectral sequence for $\mathbb{R}\oplus\mathcal{G}L^3$ to the results above.

In this subsection we describe the $\operatorname{Bss}$ for the algebra $\mathcal{G}L^4$ generated by the elements $e_1$, $e_2$, $e_3$, $e_4$ with the relations $[e_i,e_j]= (j-i) e_{i+j}$ for $i+j\leqslant 4$ and $[e_i,e_j]=0$ for $i+j>4$.

The cohomology $H^*(\mathcal{G}L^4)$ is well known and can easily be calculated using the Chevalley–Eilenberg complex. It has the following basis elements:

$$ \begin{equation*} \begin{alignedat}{2} &k=0: &&\quad 1;\\ &k=1: &&\quad \omega_1,\ \ \omega_2;\\ &k=2: &&\quad \omega_3\wedge \omega_2,\ \ \omega_4\wedge \omega_1;\\ &k=3: &&\quad \omega_4\wedge \omega_3\wedge \omega_1,\ \ \omega_4\wedge \omega_3\wedge \omega_2;\\ &k=4: &&\quad \omega_4\wedge\omega_3\wedge \omega_2\wedge \omega_1. \end{alignedat} \end{equation*} \notag $$
It is easy to see that, except for those products of elements of complementary dimensions that are non-trivial by Poincaré duality, all other products in $H^*(\mathcal{G}L^4)$ are trivial.

All three-dimensional classes and the two-dimensional class $\omega_3\wedge \omega_2$ are realised by non-trivial triple Massey products:

$$ \begin{equation*} \begin{aligned} \, \omega_3\wedge \omega_2 &= \langle - \omega_1,\omega_2,\omega_2\rangle, \\ \omega_4\wedge\omega_3\wedge\omega_2 &= \langle \omega_2,\omega_2,\omega_4\wedge\omega_1\rangle, \\ \omega_4\wedge\omega_3\wedge\omega_1 &= \biggl\langle\omega_1,\frac{1}{2}\omega_2,\omega_4\wedge\omega_1\biggr\rangle. \end{aligned} \end{equation*} \notag $$

The first equality was verified in § 7.2. To check the second equality, set $u=0$ and $v=\omega_3\wedge\omega_4$. Then

$$ \begin{equation*} du =0=\overline{\omega}_2\wedge\omega_2\quad\text{and}\quad dv= \omega_2\wedge\omega_1\wedge\omega_4= \overline{\omega}_2\wedge\omega_4\wedge\omega_1. \end{equation*} \notag $$
This implies that
$$ \begin{equation*} \overline{u}\wedge \omega_4\wedge\omega_1 + \overline{\omega}_2\wedge v= -\omega_2\wedge\omega_3\wedge\omega_4=\omega_4\wedge\omega_3\wedge\omega_2, \end{equation*} \notag $$
proving the second equality. For the third equality, set $u=\frac{1}{2}\omega_3$ and $v=\frac{1}{2}\omega_3\wedge\omega_4$. Then
$$ \begin{equation*} du=\frac{1}{2}\omega_2\wedge\omega_1=\overline{\omega}_1\wedge \frac{1}{2}\omega_2\quad\text{and}\quad dv=\frac{1}{2}\omega_2\wedge\omega_1\wedge\omega_4= \frac{1}{2}\overline{\omega}_2\wedge\omega_4\wedge\omega_1. \end{equation*} \notag $$
Therefore,
$$ \begin{equation*} \overline{u}\wedge \omega_4\wedge\omega_1 + \overline{\omega}_1\wedge v= -\frac{1}{2}\omega_3\wedge\omega_4\wedge\omega_1-\frac{1}{2}\omega_1\wedge \omega_3\wedge\omega_4=\omega_4\wedge\omega_3\wedge\omega_1, \end{equation*} \notag $$
proving the third equality.

The class $\omega_4\wedge \omega_1$ is realised by a non-trivial quadruple Massey product:

$$ \begin{equation*} \omega_4\wedge\omega_1=\langle2\omega_2,\omega_1,\omega_1,-\omega_1 \rangle. \end{equation*} \notag $$
Indeed, choose the three elements, $a_1=-2\omega_3$, $a_2=0$, and $a_3=0$, whose differentials are the products of adjacent elements of the sequence $2\omega_2,\omega_1,\omega_1,-\omega_1$: $da_1=-2\omega_2\wedge\omega_1=\overline{2\omega}_2\wedge \omega_1$, $da_2=da_3= 0$. The Massey products of three consecutive elements of the original sequence are also trivial: we set $b_1=\omega_4$ and $b_2=0$. Then
$$ \begin{equation*} d b_1=2\omega_3\wedge\omega_1=\overline{a}_1\wedge \omega_1+ \overline{2\omega}_2\wedge a_2 \quad\text{and}\quad d b_2=0=\overline{a}_2\wedge (-\omega_1)+ \overline {\omega}_1\wedge a_3 \end{equation*} \notag $$
and the quadruple Massey product is represented by the cycle
$$ \begin{equation*} \overline{b}_1\wedge (-\omega_1)+\overline{a}_1\wedge a_3 + \overline{2\omega}_2\wedge b_2=\omega_4\wedge\omega_1. \end{equation*} \notag $$

Let us verify that the Massey product $\langle2\omega_2,\omega_1,\omega_1,-\omega_1 \rangle$ is non-trivial. This can be done conveniently using the notation of § 6.1.2. Let $x(1,1)=2\omega_2$, $x(2,2)=x(3,3)=\omega_1$, $x(4,4)=-\omega_1$. Then the element $x(1,2)$ is defined uniquely and is equal to $-2\omega_3$, and the elements $x(2,3)=x$, $x(3,4)=y$ can independently take one of the three value $\pm\omega_2$ and $0$. In this case $x(1,3)=\omega_4$ and $x(2,4)=\alpha\omega_3$, where $\alpha$ is a coefficient depending on the choice of $x$ and $y$. We obtain a defining system that gives a cocycle $\omega_4\wedge\omega_1 + \beta \omega_3\wedge\omega_2$, where $\beta$ is a coefficient depending on $x$ and $y$. It is easy to see that this cocycle is not cohomologous to zero for any $x$ and $y$.

Now we describe the $\operatorname{Bss}$ for the universal enveloping algebra $U\mathcal{G}L^4$. The basis of $U \mathcal{G}L^4$ consists of monomials $e_1^a e_2^b e_3^c e_4^d$. We put

$$ \begin{equation*} p(e_1^a e_2^b e_3^ce_4^d)=a+b+2c+3d. \end{equation*} \notag $$

Lemma 9. (a) An element $x=e_1^a e_2^b e_3^ce_4^d$ for which $p(x)=p$ belongs to $N_p$ and does not belong to $N_{p-1}$.

(b) Furthermore, $N_{p }=N_{p-1} \oplus \mathbb{Q}\langle e_1^a e_2^b e_3^ce_4^d\colon a+b+2c+3d=p\rangle$.

Proof.This is similar to the proof of Theorem 7. $\Box$

Corollary 6. The generating series of the sequence $k_p=\dim N_p/N_{p-1}$ is

$$ \begin{equation*} \sum_{p\geqslant 0}k_p x^p=\frac{1}{(1-x)^2(1-x^2)(1-x^3)}\,. \end{equation*} \notag $$

Now we consider the complex $(U\mathfrak{g})^*\otimes\Lambda\mathfrak{g}^*$ dual to $U\mathfrak{g} \otimes \Lambda \mathfrak{g}$. We denote the element of $\mathfrak{g}^*$ dual to $e_i$ by $\omega_i$. Then $d\omega_k$ is given by (29), in particular, $d\omega_1=d\omega_2=0$, $d\omega_3=\omega_2\wedge\omega_1$, and $d\omega_4=2 \omega_3\wedge\omega_1$. We also denote by $(a,b,c,d)$ the element dual to the monomial $e_1^a e_2^be_3^ce_4^d$. Further in this section we use the notation $\mathfrak{g}=\mathcal{G}L^4$.

For any monomial $(a,b,c,d) \otimes \omega$, where $\omega \in \Lambda \mathfrak{g}^*$, the differential $d_{\rm CE}$ of the complex $(U\mathfrak{g})^* \otimes \Lambda \mathfrak{g}^*$ is given by

$$ \begin{equation} \begin{aligned} \, \notag d_{\rm CE}((a,b,c,d)\otimes \omega)&=(a-1,b,c,d)\otimes \omega_1\wedge\omega +(a,b-1,c,d) \otimes\omega_2\wedge \omega \\ \notag &\qquad+(a,b,c-1,d) \otimes\omega_3 \wedge \omega+(a,b,c,d-1)\otimes\omega_4\wedge \omega \\ \notag &\qquad-(b+1) (a,b+1,c-1,d)^* \otimes \omega_1\wedge\omega \\ &\qquad-2(c+1) (a,b,c+1,d-1)^* \otimes \omega_1\wedge\omega +(a,b,c,d) \otimes d\omega. \end{aligned} \end{equation} \tag{35} $$
If some of the exponents $a$, $b$, $c$, and $d$ are zero, then the corresponding terms on the right-hand side are omitted.

We collect together the properties of the differentials in the trigraded Buchstaber spectral sequence for the Lie algebra $\mathfrak{g}=\mathcal{G}L^4$.

First of all, we note that in the term $E_1^{*,*,*}$ only the groups $E_1^{p,q,*}$ on the five diagonals $-4\leqslant p+q\leqslant 0$ can be non-trivial. Moreover,

$$ \begin{equation*} \begin{alignedat}{2} E_1^{p,-p,*}&=N_p/N_{p-1},&\qquad \dim E_1^{p,-p,*}&=k_p, \\ E_1^{p,-p-1,*}&=N_p/N_{p-1}\otimes H^1,&\qquad \dim E_1^{p,-p-1,*}&=2k_p, \\ E_1^{p,-p-2,*}&=N_p/N_{p-1}\otimes H^2,&\qquad \dim E_1^{p,-p-2,*}&=2k_p, \\ E_1^{p,-p-3,*}&=N_p/N_{p-1}\otimes H^3,&\qquad \dim E_1^{p,-p-4,*}&=k_p. \end{alignedat} \end{equation*} \notag $$
Here we use the notation $H^k=H^k(\mathcal{G}L^4)$.

Statement 4. (a) The differential $d_1\colon N_p/N_{p-1}\to N_{p-1}/N_{p-2}\otimes H^1$ is injective. For $p=1$ it is an isomorphism.

(b) The differential $d_1\colon N_{p-1}/N_{p-2}\otimes H^1\to N_{p-2}/N_{p-3}\otimes H^2$ is zero.

(c) The differential $d_1\colon N_{p-2}/N_{p-3}\otimes H^2\to N_{p-3}/N_{p-4}\otimes H^3$ is zero.

(d) The differential $d_1\colon N_{p-3}/N_{p-4}\otimes H^3\to N_{p-4}/N_{p-5}\otimes H^4$ is surjective for $p\geqslant 3$.

Proof. Parts (a)–(c) follow from the general properties of the differential $d_1$ (see Theorem 2).

To prove part (d), note that $(a,b,c,d) \otimes \omega_4\wedge\omega_3\wedge\omega_2\wedge\omega_1$ is equal to $d_1((a,b+ 1,c,d) \otimes \omega_4\wedge\omega_3\wedge\omega_1)$. $\Box$

As a corollary, we obtain the following description of the second page $E_2^{*,*,*}$:

Theorem 13. The following statements hold:

(0) $ E_2^{p,-p,*}=0$;

(1) $\dim E_2^{p,-p-1,*}=2k_p- k_{p+1}$;

(2) $E_2^{p,-p-2,*}=E_1^{p,-p-2,*}$, $\dim E_2^{p,-p-2,*}=2k_p$;

(3) $\dim E_2^{p,-p-3,*}=2k_p- k_{p-1}$;

(4) $E_2^{p,-p-4,*}=0$.

To describe the differential $d_2$ we need to choose basis elements in $E_2^{p,-p-1,*}$ in terms of their representatives in $E_1^{p,-p-1,*}$. Any element in $E_1^{p,-p-1,*}$ is a linear combination of elements $(a,b,c,d)\otimes \omega_1$ and $(a,b,c,d)\otimes \omega_2$, where $a+b+2c+3d=p$.

Lemma 10. Independent elements of $E_2^{p,-p-1,*}$ are elements represented in $E_1^{p,-p-1,*}$ by cycles of the following types:

(i) $(a,b,c,d)\otimes \omega_1$, where $b\geqslant 2$;

(ii) $(a,0,c,d)\otimes \omega_1$, where $c\geqslant 2$;

(iii) $(a,0,0,d)\otimes \omega_1$, where $d\geqslant 1$.

Proof. Elements of $E_2^{p,-p-1,*}$ are defined by their representing cycles in $E_1^{p,-p-1,*}$ up to addition of elements of the form $d_1(A,B,C,D)$, where $(A,B,C,D)\in E_1^{p+1,-p-1,*}$.

It is easy to see that $d_1(a,b+1,c,d)=(a,b,c,d)\otimes\omega_2+u\otimes\omega_1$, where $u\in N_p/N_{p-1}$. Hence, the equality $(a,b,c,d)\otimes\omega_2=-u\otimes \omega_1$ holds in $E_2^{p,-p-1,*}$, and therefore any element of $E_2^{p,-p-1,*}$ can be represented by a cycle in $E_1^{p,-p-1,*}$ that is a linear combination of elements $(a,b,c,d)\otimes \omega_1$.

Now observe that

$$ \begin{equation*} d_1(a,0,c+1,d)=\bigl((a-1,0,b+1,d)-2(c+2)(a,0,c+2,d-1)- (a,1,c,d)\bigr)\otimes \omega_1. \end{equation*} \notag $$
Therefore, the following identity holds in $E_2^{p,-p-1,*}$:
$$ \begin{equation*} (a,1,c,d)\otimes \omega_1=(a-1,0,b+1,d)\otimes \omega_1- 2(c+2)(a,0,c+2,d-1)\otimes \omega_1. \end{equation*} \notag $$

Furthermore, $d_1(a,0,0,d+1)=\bigl((a-1,0,0,d+1)-3(a,0,1,d)\bigr)\otimes\omega_1$. Hence the following identity holds in $E_2^{p,-p-1,*}$:

$$ \begin{equation*} (a,0,1,d)\otimes \omega_1=\frac{1}{3}(a-1,0,0,d+1)\otimes \omega_1. \end{equation*} \notag $$

Finally, an element of the form $(a,0,0,0)\otimes \omega_1$ is equal to $d_1(a+1,0,0,0)$.

It remains to show that the number of elements $(a,b,c,d)\otimes \omega_1$ of the form described in the statement of the lemma, where $a+b+2b+3c=p$, is exactly $2 k_p-k_{p+1}$.

To do this we calculate the numbers of elements of different types using generating series. Namely, the generating series of filtration $p$ for the number of elements

$\bullet$ of the form $(a,b,c,d)$, where $b\geqslant 2$, is equal to $x^2/((1-x)^2(1-x^2)(1-x^3))$;

$\bullet$ of the form $(a,0,c,d)$, where $c\geqslant 2$, is equal to $x^4/((1-x)(1-x^2)(1-x^3))$;

$\bullet$ of the form $(a,0,0,d)$, where $d\geqslant 1$, is equal to $x^3/((1-x)(1-x^3))$.

Now the statement follows from the easily verifiable identity

$$ \begin{equation*} \begin{aligned} \, &\frac{x^2}{(1-x)^2(1-x^2)(1-x^3)}+\frac{x^4}{(1-x)(1-x^2)(1-x^3)}+ \frac{x^3}{(1-x)(1-x^3)} \\ &\qquad=\frac{2-1/x}{(1-x)^2(1-x^2)(1-x^3)}+\frac{1}{x}\,. \end{aligned} \end{equation*} \notag $$
Here the first summand on the right-hand side is the generating series for the sequence $2k_p-k_{p+1}$ for $p\geqslant 0$, and the second summand $1/x$ is needed to remove from the sum the meaningless term for $p=-1$. $\Box$

The next step is to compute the differentials $d_2$ and $d_3$ at the elements from Lemma 10.

Lemma 11. (a) The differential $d_2\colon E_2^{p,-p-1,*}\to E_2^{p-2,-p,*}= N_{p-2}/N_{p-3}\otimes H^2$ is non-trivial on elements $(a,b,c,d)\otimes\omega_1$ for $b\geqslant 2$; more precisely,

$$ \begin{equation} d_2 \bigl((a,b,c,d)\otimes\omega_1\bigr)= (a,b-2,c,d)\otimes\omega_3\wedge\omega_2. \end{equation} \tag{36} $$

(b) The differential $d_2\colon E_2^{p,-p-1,*}\to E_2^{p-2,-p,*}= N_{p-2}/N_{p-3}\otimes H^2$ is trivial on elements $(a,0,c,d)\otimes\omega_1$ for $c\geqslant 2$ and on elements $(a,0,0,d)\otimes\omega_1$ for $d\geqslant 1$.

Proof. This goes by direct calculation. $\Box$

Corollary 7. The classes represented by the elements $(a,0,c,d)\otimes\omega_1$ with $c\geqslant 2$ and $(a,0,0,d)\otimes\omega_1$ with $d\geqslant 1$ form a basis in $E_3^{-p,-p-1,*}$

Lemma 12. The differential

$$ \begin{equation*} d_2\colon E_2^{p,-p-2,*}=N_p/N_{p-1}\otimes H^2 \to E_2^{p-2,-p-1,*}=N_{p-2}/N_{p-3}\otimes H^3 \end{equation*} \notag $$
is surjective. In particular, $E_3^{p-2,-p-1,*}=0$.

Proof. Let $M$ be the submodule of $E_2^{p,-p-2,*}=N_p/N_{p-1}\otimes H^2$ generated by the elements $(a,b,c,d)\otimes \omega_4\wedge\omega_1$ for $b\geqslant 2$ and $(a,0,c,d)\otimes \omega_4\wedge\omega_1$ for $c\geqslant 1$. It follows from Lemma 11 that the intersection of $M$ with the image of $d_2$ is zero. A direct calculation shows that
$$ \begin{equation*} \begin{aligned} \, d_2 \bigl((a,b,c,d)\otimes\omega_4\wedge\omega_1\bigr) &= - (a,b-2,c,d)\otimes\omega_4\wedge\omega_3\wedge\omega_2 + u\otimes \omega_4\wedge\omega_3\wedge\omega_1, \\ d_2 \bigl((a,0,c,d)\otimes\omega_4\wedge\omega_1\bigr) &= -(a,0,c-1,d)\otimes\omega_4\wedge\omega_3\wedge\omega_1, \end{aligned} \end{equation*} \notag $$
where $u$ is some element of $N_{p-2}/N_{p-3}$. It is easy to see that the elements on the right-hand sides of these equalities are independent. Thus, we obtain an inclusion of $M$ to a submodule in $E_2^{p-2,-p-1,*}=N_{p-2}/N_{p-3}\otimes H^3$. Let us show that this is actually an isomorphism.

The dimensions of $M$ for all possible filtrations are given by the generating series

$$ \begin{equation} \frac{x^2}{(1-x)^2(1-x^2)(1-x^3)}+\frac{x^2}{(1-x)(1-x^2)(1-x^3)}\,. \end{equation} \tag{37} $$
On the other hand, by part (3) of Theorem 13 the dimensions of the $E_2^{p-2,-p-1,*}$ are given by the generating series
$$ \begin{equation*} \frac{2-x}{(1-x)^2(1-x^2)(1-x^3)}\,. \end{equation*} \notag $$
We need to show that the coefficients of this series and the series (37) coincide up to a shift of the dimensions by $2$. This follows from the identity
$$ \begin{equation*} \begin{aligned} \, \frac{x^2}{(1-x)^2(1-x^2)(1-x^3)}&+\frac{x^2}{(1-x)(1-x^2)(1-x^3)} \\ &\qquad=x^2\,\frac{2-x}{(1-x)^2(1-x^2)(1-x^3)}\,. \ \ \Box \end{aligned} \end{equation*} \notag $$

The previous arguments imply the following facts about $E_3^{*,*,*}$. The non-zero groups are positioned on two diagonals, $E_3^{p,-p-1,*}$ and $E_3^{p,-p-2,*}$. The elements $(a,0,c,d)\otimes\omega_1$ for $c\geqslant 2$ and $(a,0,0,d)\otimes\omega_1$ for $d\geqslant 1$ represent a basis of $E_3^{p,-p-1,*}$ (see Lemma 11). The generating series for $\dim E_3^{p,-p-1,*}$ is therefore given by

$$ \begin{equation*} A(x)=\frac{x^4 }{(1-x)(1-x^2)(1-x^3)}+ \frac{x^3}{(1-x)(1-x^3)}\,. \end{equation*} \notag $$

On the other hand, Lemma 12 does not provide explicit representatives for a basis of $E_3^{p,-p-2,*}$, but we can calculate its dimension. To do this, recall that

$$ \begin{equation*} E_2^{p,-p-2,*}=N_p/N_{p-1}\otimes \omega_3\wedge\omega_2 \oplus N_p/N_{p-1}\otimes \omega_4\wedge\omega_1. \end{equation*} \notag $$
In this direct sum the first term coincides with the image of $d_2$ (see Lemma 11), and the second term is mapped by $d_2$ onto $E_2^{p-2,-p-1,*}$ (see Lemma 12). Hence,
$$ \begin{equation*} \dim E_3^{p,-p-2,*}=\dim N_p/N_{p-1} - \dim E_2^{p-2,-p-1,*}=k_p - (2 k_{p-2} - k_{p-3}). \end{equation*} \notag $$
The corresponding generating series is
$$ \begin{equation*} B(x)=\frac{1- 2x^2 + x^3 }{(1-x)^2 (1-x^2)(1-x^3)}\,. \end{equation*} \notag $$

Lemma 13. The dimensions of the non-zero groups $E_3^{*,*,*}$ connected by the differential $d_3$ are equal:

$$ \begin{equation*} \dim E_3^{p,-p-1,*}=\dim E_3^{p-3,-p+1}. \end{equation*} \notag $$

Proof. This follows from the equality of the generating series with a dimension shift by $3$:
$$ \begin{equation*} A(x)= x^3\,B(x). \Box \end{equation*} \notag $$

Theorem 14. The differentials $d_3\colon E_3^{p,-p-1,*} \to E_3^{p-3,-p+1}$ are isomorphisms. In particular, $E_4^{*,*,*}=E_\infty^{*,*,*}$.

Proof. Consider all $p\geqslant 3$ consecutively. For $p=3$ the differential $d_3$ is injective for dimensional reasons, so it is an isomorphism by Lemma 13. Then consider $d_3$ for $p= 4$, and so on. $\Box$

Remark 7. This theorem can also be derived from Theorem 5.

We also present explicit formulae for the values of $d_3$ at generators of $E_3^{p,-p-2,*}$, although we do not use them.

Lemma 14. The differential $d_3\colon E_3^{p,-p-1,*}\to E_3^{p-2,-p,*}= N_{p-3}/N_{p-4}\otimes H^2$ is non-trivial at elements $(a,0,c,d)\otimes\omega_1$ for $c\geqslant 2$ and at elements $(a,0,0,d)\otimes\omega_1$ for $d\geqslant 1$. More precisely,

$$ \begin{equation} d_3\bigl((a,0,c,d)\otimes\omega_1\bigr) =-\frac{1}{2}(a,1,c-2,d)\otimes \omega_4\wedge\omega_1+\frac{1}{2}(a-1,0,c-1,d)\otimes\omega_4\wedge\omega_1 \end{equation} \tag{38} $$
and
$$ \begin{equation} d_3 \bigl((a,0,0,d)\otimes\omega_1\bigr)= (a,0,0,d-1)\otimes\omega_4\wedge\omega_1. \end{equation} \tag{39} $$

7.4. Realisation of two-dimensional cohomology classes

The realisation of two-dimensional cohomology classes of manifolds $M^3$ and $M^4$ was discussed above. Here we discuss the realisation of generators of the two-dimensional cohomology of nilmanifolds $M^n$ for $n\geqslant 5$. It is well known that the second Betti number of a nilmanifold $M^n$ for $n\geqslant 5$ is $3$ (see [71]). Two generators are the classes $[\omega_3\wedge\omega_2]$ and $[\omega_5\wedge\omega_2 - 3\omega_4\wedge\omega_3]$, which are stable in the tower (33), that is, these classes are mapped to the same classes under the homomorphism in the cohomology induced by the bundle projection $p_n\colon M^{n+1}\to M^n$ for $n\geqslant 5$. Note that precisely these two classes are present in the two-dimensional cohomology of the infinite-dimensional Lie algebra $L_1$. Its completion in the topology defined by the grading is the inverse limit of the algebras $\mathcal{G}L^n$. The realisation of these classes by Massey products or by the images of differentials in $\operatorname{Bss}$ is exactly the same as in the cohomology of the Lie algebra $L_1$ (see the end of § 5.2).

The third generator of $H^2(M^n)$ is the class

$$ \begin{equation*} \Omega_n=\sum_{i=1}^{[n/2]} (n-2i)\omega_{n+1-i}\wedge \omega_i. \end{equation*} \notag $$
It is easy to see that $p_n^*(\Omega_n)$ is zero in the cohomology of $M^{n+1}$. Indeed, there is a 1-form $\omega_{n+1}$ on $M^{n+1}$ such that $d\omega_{n+1}=\Omega_n$.

For even $n=2k$ this class is very important: the form $\Omega_{2k}$ defines a symplectic structure on the manifold $M^{2k}$. Moreover, $p_n\colon M^{n+1}\to M^n$ is a principal bundle with fibre $S^1$ for any $n$. The 1-form $\omega_{n+1}$ is the connection form of this bundle, and the form $\Omega_n$ on the base $M^n$ is its curvature form: $d\omega_{n+1}=p_n^*(\Omega_{n})$.

As shown in [14], the two-dimensional class $[\Omega_{2k}]\in H^2(M^{2k})$ represents the $2k$-fold Massey product

$$ \begin{equation} \biggl\langle \omega_1, (k-1)\omega_1, (k-2)\omega_1,\dots, 2\omega_1, \begin{pmatrix}\omega_1 & \omega_2\end{pmatrix}, \begin{pmatrix}0\\-\omega_1\end{pmatrix},-2\omega_1,\dots,-(k-1)\omega_1, -k \omega_1\biggr\rangle \end{equation} \tag{40} $$
with the defining system
$$ \begin{equation*} \begin{pmatrix} \omega_1 & \omega_2 &\dots & \omega_{2k-1}& \omega_{2k} & \\ & (k-1)\omega_1 & (k-1)\omega_2 &\dots & (k-1)\omega_{2k-1}& (k-1)\omega_{2k} \\ &&\dots&\dots&\dots&\dots \\ &&&& -(k-1)\omega_1& -(k-1)\omega_2 \\ &&&&& -k\omega_1 \end{pmatrix}. \end{equation*} \notag $$
Moreover, it was shown in [14] that $\Omega_{2k}$ cannot be a representative of any linear combination of Massey products of fewer arguments. This implies that the above representation of $[\Omega_{2n}]$ as a $2n$-fold Massey product is non-trivial, that is, it does not contain zero. Indeed, otherwise the indeterminacy of the Massey product (40) would contain $[\Omega_{2n}]$. On the other hand, the indeterminacy of an $2n$-fold matric Massey product is contained in the union of some $(2n-1)$-fold matric Massey products (see § 6.1.3), which implies that $[\Omega_{2n}]$ is a representative of an $(2n-1)$-fold matric Massey product.

The result of [14] can be extended to odd-dimensional manifolds $M^n$. Namely, for $n=2k+1$ the class $[\Omega_{2k+1}]\in H^2(M^{2k+1})$ belongs to the $(2k+1)$-fold matric Massey product

$$ \begin{equation} \biggl\langle \omega_1, k\omega_1, (k-1)\omega_1,\dots, 2\omega_1, \begin{pmatrix}\omega_1 & \omega_2\end{pmatrix}, \begin{pmatrix}0\\-\omega_1\end{pmatrix},-2\omega_1,\dots, -(k-1)\omega_1,-k \omega_1\biggr\rangle \end{equation} \tag{41} $$
with the defining system
$$ \begin{equation*} \begin{pmatrix} \omega_1 & \omega_2 &\ldots & \omega_{2k}& \omega_{2k+1} & \\ & k\omega_1 & k \omega_2 &\ldots & k\omega_{2k}& k\omega_{2k+1} \\ &&\cdots&\cdots&\cdots&\cdots \\ &&&& -(k-1)\omega_1& -(k-1)\omega_2 \\ &&&&& -k\omega_1 \end{pmatrix} \end{equation*} \notag $$
and it is not a representative of a linear combination of Massey products with fewer arguments. The proof of non-representability is the same as in the case $n=2k$, it uses bigrading and the filtration of the complex $\Lambda(\omega_1,\dots,\omega_n)$ by the subcomplexes $\Lambda(\omega_1,\dots,\omega_p)$, $p\leqslant n$. As above, this immediately implies the non-triviality of the Massey product (41).

Statement 5. The cohomology class of $\Omega_n$ belongs to $\Phi^{n-1}H^2(M^{2n})$ and does not belong to $\Phi^{n-2}H^2(M^{2n})$.

Proof. If the cohomology class $[\Omega_n]\in H^2(M^n)$ had a filtration $\Phi^p$ with $p< n-1$, then it would be represented by a $(p+1)$-fold Massey product. It follows from the above that this is impossible, since $p+1<n$. Hence, the class $[\Omega_n]\in H^2(M^n)$ is of filtration $\Phi^p$, where $p\geqslant n-1$.

By Theorem 5 the differentials $d_p$ in the $\operatorname{Bss}$ of the algebra $U\mathcal{G}L^n$ are zero for $p>n-1$. Consequently, $[\Omega_n]$ has filtration precisely $n-1$. $\Box$

Now note that $[\Omega_n]$ has the third grading $-(2n+2)$ in the $\operatorname{Bss}$, that is, $[\Omega_n]$ belongs to $E^{0,-2,-2n-2}_{1}$ and survives to $E^{0,-2,-2n-2}_{n-1}$. The differentials of $\operatorname{Bss}$ are homogeneous with respect to the third grading, so the question arises of which element is mapped to the class $[\Omega_n]$ under the action of the differential $d_{n-1}\colon E^{n-1,-n-2,-2n-2}_{n-1}\to E^{0,-2,-2n-2}_{n-1}$ in the $\operatorname{Bss}$ of the Lie algebra $\mathcal{G}L^n$.

Theorem 15. For $n\geqslant 2$ there is only one class in the group $E^{n-1,-n,-2n-2}_{2}$. It is represented by $[\theta_n]=(n-1)[e_n^*\otimes \omega_1]$. This class survives to $E^{n-1,-n,-2n-2}_{n-1}$, and $d_{n-1}([\theta_n])$ is a generator of $E^{0,-2,-2n-2}_{n-1}$.

Proof. It is easy to check that the group $E_1^{n-1,-n,-2n-2}$ has a basis of the following elements:
$$ \begin{equation} (e_1^k e_{n-k})^*\otimes \omega_1,\quad \text{where}\ \ 0\leqslant k\leqslant n-2,\quad \text{and}\quad e_1^{n-1}\otimes \omega_2, \end{equation} \tag{42} $$
hence, $\dim E_1^{n-1,-n,-2n-2}=n$.

On the other hand, there are only the following elements of the third grading $-2n-2$ in $E_1^{n,-n,-2n-2}= N_{n}/N_{n-1}$:

$$ \begin{equation*} (e_1^ke_{n+1-k})^*,\quad \text{where}\ \ 1\leqslant k\leqslant n-1, \end{equation*} \notag $$
which implies that $\dim E_1^{n,-n,-2n-2}=n-1$.

Now recall that by Theorem 2 (b) the differential $d_1\colon E_1^{n-1,-n,-2n-2} \to E_1^{n-2,-n,-2n-2}$ is expressed via the product

$$ \begin{equation*} H^1(\mathcal{G}L^n) \otimes H^1(\mathcal{G}L^n) \to H^2(\mathcal{G}L^n), \end{equation*} \notag $$
which is trivial. Indeed, $H^1(\mathcal{G}L^n)$ is generated by the two classes $[\omega_1]$ and $[\omega_2]$, where $\omega_2\wedge\omega_1=d\omega_3$. Thus,
$$ \begin{equation*} \ker d_1 \colon E_1^{n-1,-n,-2n-2} \to E_1^{n-2,-n,-2n-2} \end{equation*} \notag $$
is the whole space $E_1^{n-1,-n,-2n-2}$.

By Theorem 2 (b), the differential $d_1\colon E_1^{n,-n,-2n-2}\to E_1^{n-1,-n,-2n-2}$ is injective, so $\dim E_2^{n-1,-n,-2n-2}=1$.

Calculating the differential $d_1$ at the elements $(e_1^ke_{n+1-k})^*$ for $1\leqslant k\leqslant n-2$ we obtain

$$ \begin{equation*} [(e_{1}^{k-1} e_{n+1-k})^*\otimes \omega_1]= \lambda_k [(e_{1}^{k} e_{n-k})^*\otimes \omega_1] \quad\text{in} \ \ E_2^{n-1,-n,-2n-2}, \end{equation*} \notag $$
where $\lambda_k\ne 0$. Finally,
$$ \begin{equation*} d_1 (e_1^ne_2)^*=((e_1)^{n-1})^*\otimes \omega_2 + ((e_1)^{n-2}e_2)^*\otimes \omega_1. \end{equation*} \notag $$
This means that any class in (42) can be taken as a generator of the one-dimensional space $E_2^{n-1,-n,-2n-2}$. We set
$$ \begin{equation*} [\theta_n]=(n-1)[e_n^*\otimes \omega_1]. \end{equation*} \notag $$

Since $[\Omega_n]$ survives to $E_{n-1}^{0,-2,-2n-2}$, the differential

$$ \begin{equation*} d_{n-1}\colon E_{n-1}^{n-1,-n,-2n-2} \to E_{n-1}^{0,-2,-2n-2} \end{equation*} \notag $$
is an isomorphism. $\Box$

Now for small $n$ we can give an explicit formula for the cochains $\theta_n$ representing the classes $(n-1)[e_n^*\otimes \omega_1]$, mapped to the classes $[\Omega_n]$ by the differentials of $\operatorname{Bss}$ for the algebra $U\mathcal{G}L^n$. The cochains $\theta_4$ and $\theta_5$ were obtained directly, while the cochains $\theta_6$ and $\theta_7$ were obtained by reducing the problem to a system of linear equations, which was solved using a software package.

The case $n=4$. The form $\Omega_4=3\omega_4\wedge\omega_1+\omega_3\wedge\omega_2$ is a linear combination of the cocycles $\omega_4\wedge\omega_1$ and $\omega_3\wedge\omega_2$, representing classes of filtration 3 and 2, respectively. The class $[\Omega_4 ]$ itself is equal to $d_3(3e_4^*\otimes \omega_1)$. Consider the cohain

$$ \begin{equation*} \theta_4=3 e_4^*\otimes \omega_1+e_3^*\otimes \omega_2- (e_1e_2)^*\otimes \omega_2. \end{equation*} \notag $$
It defines the same class as $3e_4^*\otimes \omega_1$ in the first term of $\operatorname{Bss}$ or, more precisely, in the group $E_1^{3,-4,*}$. On the other hand, one can check that $d_{\rm CE}(\theta_4)= \Omega_4$, so the class of the form $3e_4^*\otimes \omega_1$ belongs to the kernels of the differentials $d_1$ and $d_2$, and is mapped to $[\Omega_4]$ by the differential $d_3$.

The case $n=5$. The class $[\Omega_5]=[4\omega_5\wedge\omega_1+ 2\omega_4\wedge\omega_2]$ is equal to $d_4(4 e_5^*\otimes \omega_1)$. The cochain is

$$ \begin{equation*} \theta_5=4 e_5^*\otimes \omega_1 + 2e_4^*\otimes \omega_2. \end{equation*} \notag $$
It defines the same class as $4 e_5^*\otimes \omega_1$ in the group $E_1^{4,-5,*}$. One can check that $d_{\rm CE}(\theta_5)=\Omega_5$, so the class of the form $4 e_5^*\otimes \omega_1$ belongs to the kernels of the differentials $d_1$, $d_2$, and $d_3$, and is mapped to $[\Omega_5]$ by the differential $d_4$.

The case $n=6$. The class $[\Omega_6]=[5\omega_6\wedge\omega_1+3\omega_5\wedge\omega_2+ \omega_4\wedge\omega_3]$ is equal to $d_5(5 e_6^*\otimes \omega_1)$. The cochain is

$$ \begin{equation*} \theta_6=5 e_6^*\otimes \omega_1 + 3e_5^*\otimes \omega_2+ e_4^*\otimes \omega_3 - 2(e_3^2)^*\otimes \omega_1. \end{equation*} \notag $$
It defines the same class as $5 e_6^*\otimes \omega_1$ in the group $E_1^{5,-6,*}$. One can check that $d_{\rm CE}(\theta_6)=\Omega_6 $, so the class of the form $5 e_6^*\otimes \omega_1$ belongs to the kernels of $d_1$, $d_2$, $d_3$, and $d_4$, and is mapped to $[\Omega_6]$ by $d_5$.

The case $n=7$. The class $[\Omega_7]=[6\omega_7\wedge\omega_1 + 4\omega_6\wedge\omega_2 + 2\omega_5\wedge\omega_3]$ is equal to $d_6( 6 e_7^*\otimes \omega_1)$. The cochain is

$$ \begin{equation*} \theta_7=6 e_7^*\otimes \omega_1 + 4e_6^*\otimes \omega_2+ 2e_4^*\otimes \omega_4 + 2(e_3^2)^*\otimes \omega_2 -2e_3^* \wedge\omega_5 + 2(e_3e_4)^*\otimes \omega_1+ 2(e_2e_5)^*\omega_1. \end{equation*} \notag $$
It defines the same class as $5e_7^*\otimes \omega_1$ in the group $E_1^{6,-7,*}$. One can check that $d_{\rm CE}(\theta_6)=\Omega_7$, so the class of the form $6e_7^*\otimes\omega_1$ belongs to the kernels of $d_1$, $d_2$, $d_3$, $d_4$, and $d_5$, and is mapped to $[\Omega_7]$ by $d_6$.

8. Heisenberg nilmanifolds $M_H^{2n+1}$

In this section we collect results on the real cohomology of the Heisenberg nilmanifolds and present solutions to problems (1), (2) and (3) for the Lie algebras $\mathcal{GH}^{2n+1}$.

8.1. Homogeneous spaces of the Heisenberg group

Consider the group ${\mathcal H}^{2n+1}$ of matrices

$$ \begin{equation} \begin{pmatrix} 1 & x_1 & x_2 & \dots & x_n & z \\ 0 & 1 & 0 & \dots & 0 & y_1 \\ & \ddots & \ddots & \ddots &\vdots &\vdots \\ & & \ddots & 1 & 0& y_{n-1} \\ & {\large\textbf{0}} & & 0 & 1& y_n \\ && & & 0 & 1 \end{pmatrix} \end{equation} \tag{43} $$
with real $x_j$, $y_j$, and $z$. As a space, it is homeomorphic to $\mathbb{R}^{2n+1}$. This group has a cocompact lattice $\Gamma_H^{2n+1}$ consisting of matrices with integer $x_j$, $y_j$, and $z$.

The Lie algebra $\mathcal{GH}^{2n+1}$ consists of the matrices

$$ \begin{equation*} A=\begin{pmatrix} 0 & x_1 & x_2 & \cdots & x_n & z \\ 0 & 0 & 0 & \cdots & 0 & y_1 \\ &\ddots &\ddots &\ddots &\vdots & \vdots \\ & &\ddots & 0 & 0& y_{n-1} \\ & {\large\textbf{0}} && 0 & 0& y_n \\ &&& & 0 & 0 \end{pmatrix} \end{equation*} \notag $$
with real $x_j$, $y_j$ and $z$. There is a basis $e_{0},e_{\pm 1},\dots,e_{\pm n}$ in $\mathcal{GH}^{2n+1}$ such that
$$ \begin{equation*} A=x_1e_{-1}+\cdots+x_n e_{-n} + y_1 e_{1} + \cdots+y_n e_{n } + z e_{0}= \sum_{j=1}^n x_j e_{-j} + \sum_{j=1}^n y_j e_j + z e_0. \end{equation*} \notag $$
A direct check shows that in $\mathcal{GH}^{2n+1}$ all commutators of basis elements vanish, except for $[e_{-j},e_{j}]=e_{0}$, $j=1,\dots,n$. Thus, the Lie algebra $\mathcal{GH}^{2n+1}$ is nilpotent and its structure constants are integers.

The group ${\mathcal H}^{2n+1}$ for $n\geqslant 3$ is known as the generalised Heisenberg group. Its Lie algebra is obtained by replacing the canonical Poisson bracket on the space $\mathbb{R}^{2n}$ by a commutator, and the unity in the algebra of functions on $\mathbb{R}^n$ by the element $e_{0}$, which plays the role of the Planck constant.

Thus we obtain a series of nilmanifolds $M_H^{2n+1}={\mathcal H}^{2n+1} /\Gamma_H^{2n+1}$ with $M_H^3=M^3$. There is a sequence of inclusions

$$ \begin{equation*} S^1=M^1_H\xrightarrow{i_0}M^3_H \xrightarrow{i_1} M^5_H\xrightarrow{i_2}\cdots, \end{equation*} \notag $$
induced by the inclusions of subgroups
$$ \begin{equation*} {\mathcal H}^{1}\xrightarrow{i_0} {\mathcal H}^{3}\xrightarrow{i_1} {\mathcal H}^{5}\to\cdots. \end{equation*} \notag $$
Here the inclusion $i_n\colon{\mathcal H}^{2n+1}\xrightarrow{i_k} {\mathcal H}^{2n+3}$ is defined by
$$ \begin{equation*} \begin{pmatrix} 1 & x_1 & x_2 & \dots & x_n & z \\ 0 & 1 & 0 & \dots & 0 & y_1 \\ &\ddots&\ddots &\ddots &\vdots&\vdots \\ & & \ddots & 1 & 0& y_{n-1} \\ & {\large\textbf{0}}& & 0 & 1& y_n \\ && & & 0 & 1 \end{pmatrix} \mapsto \begin{pmatrix} 1 & x_1 & x_2 & \dots & x_n & 0& z \\ 0 & 1 & 0 & \dots & 0 & 0& y_1 \\ &\ddots&\ddots &\ddots &0&\vdots&\vdots \\ & &\ddots & 1 & 0& 0& y_{n-1} \\ & & & 0 & 1& 0& y_n \\ & {\large\textbf{0}} & & & 0& 1& 0 \\ && & & & 0& 1 \end{pmatrix}, \end{equation*} \notag $$
where the basis elements $e_{0},e_{\pm 1},\dots,e_{\pm n}$ of $\mathcal{GH}^{2n+1}$ are mapped to the same basis elements of $\mathcal{GH}^{2n+3}$.

By Nomizu’s theorem the real cohomology ring $H^*(M_H^{2n+1})$ is isomorphic to the cohomology ring $H^*(\mathcal{GH}^{2n+1})$. Let $\omega_{k}$ denote the element dual to $e_k$. Then the differential of the Chevalley–Eilenberg complex

$$ \begin{equation*} C^*_{\rm CE}(\mathcal{GH}^{2n+1})= \wedge(\omega_0,\omega_{\pm 1},\dots,\omega_{\pm n}) \end{equation*} \notag $$
is described by the relations
$$ \begin{equation*} d\omega_{\pm 1}=\cdots=d\omega_{\pm n}=0,\qquad d\omega_{0}=\sum_{k=1}^n\omega_{-k}\wedge\omega_{k}. \end{equation*} \notag $$

The form $\omega_0$ defines the structure of a contact manifold on $M_H^{2n+1}$, since the form $\omega_0\wedge (d\omega_0)^n$ is proportional to the form $\omega_0\wedge\omega_{-1}\wedge\omega_1\wedge\cdots\wedge \omega_{-n}\wedge\omega_n$ representing a generator of the group $H^{2n+1}(\mathcal{GH}^{2n+1})$.

Mapping the matrix (43) to the matrix of the same form, but with $z=0$, we obtain a locally trivial bundle

$$ \begin{equation} \pi\colon M^{2n+1}_H\to T^{2n}, \end{equation} \tag{44} $$
with fibre $S^1$.

Nomizu’s theorem implies that the projection of the bundle induces a morphism of Chevalley–Eilenberg complexes

$$ \begin{equation*} \begin{aligned} \, \pi^*\colon\Lambda(({\mathfrak t}^{2n})^*)&= \bigl(\Lambda(\omega_{\pm k}\colon 1\leqslant k\leqslant n),d_{\rm CE}=0\bigr) \to \bigl(\Lambda (\omega_0,\omega_{\pm k}\colon 1\leqslant k \leqslant n) \\ &=\Lambda((\mathcal{GH}^{2n+1})^*),d_{\rm CE}\bigr), \end{aligned} \end{equation*} \notag $$
and the differential $d_{\rm CE}$ is zero on all generators, except that
$$ \begin{equation*} d_{\rm CE}(\omega_0)=\sum_{k=1}^n\omega_{-k}\wedge\omega_k. \end{equation*} \notag $$
Here the generators are chosen so that for $k\ne 0$ we have
$$ \begin{equation*} \Lambda(({\mathfrak t}^{2n})^*) \ni \omega_k \overset{\pi^*}{\longmapsto} \omega_k\in \Lambda((\mathcal{GH}^{2n+1})^*), \end{equation*} \notag $$
where $\omega_0$ is the generator in the cohomology of the fibre $S^1$.

The spectral sequence of the bundle (44) for real cohomology coincides with the Hochschild–Serre spectral sequence of the one-dimensional ideal generated by $e_0$ in the Lie algebra $\mathcal{GH}^{2n+1}$:

$$ \begin{equation*} E_2^{p,q}=H^p(\mathfrak{t}^{2n},H^q(\Lambda(\omega_0))). \end{equation*} \notag $$
Note that the representation of the algebra $\mathfrak{t}^{2n}$ in the cohomology $H^q(\Lambda (\omega_0))$ is trivial, since the extension $\mathfrak{t}^{2n}\subset \mathcal{GH}^{2n+1}$ is central.

It is easy to see that the differential $d_2\colon H^1(S^1)=E_2^{0,1}\to E_2^{2,0}=H^2(T^{2n})$ is defined by the familiar formula:

$$ \begin{equation*} d_2(\omega_0)=\sum_{k=1}^n \omega_{-k}\wedge\omega_k. \end{equation*} \notag $$
On the other groups $E_{2}^{p,1}$ the differential $d_2$ is defined by multiplicativity.

8.2. The cohomology of the Lie algebra $\mathcal{GH}^{2n+1}$

We assign bigrading $(1,k)$ to the form $\omega_k$. Then, as in the case of the manifolds $M^n$, we obtain that the Chevalley–Eilenberg differential preserves the second grading. Hence there is a splitting of the complex $C^*_{\rm CE}(\mathcal{GH}^{2n+1})$ into a direct sum of subcomplexes, which induces a natural bigrading on the cohomology ring:

$$ \begin{equation*} H^*(\mathcal{GH}^{2n+1})= \bigoplus_{p\geqslant 0,q}H^{p,q}(\mathcal{GH}^{2n+1}). \end{equation*} \notag $$
The presence of such a bigrading allows us to expect that many results of [60] for the manifolds $M^{n}$ are also true for the manifolds $M_H^{2n+1}$.

Remark 8. A different grading of the basis elements is often used: $e_1,\dots,e_n$ are denoted by $e_{2n},\dots,e_{n+1}$, then $e_{-n},\dots,e_{-1}$ are denoted by $e_1,\dots,e_n$, and the element $e_0$ is denoted by $e_{2n+1}$. The commutation relations $[e_{-n},e_n]=e_0$ take the form $[e_i,e_{2n+1-i}]=e_{2n+1}$. Our choice of numbering of the basis elements is due to the fact that the inclusion $i_n\colon{\mathcal H}^{2n+1}\xrightarrow{i_k}{\mathcal H}^{2n+3}$ takes the generators of the algebra $\mathcal{GH}^{2n+1}$ to the corresponding generators of $\mathcal{GH}^{2n+3}$.

The manifolds $M_H^{2n+1}$ are compact and orientable, and so they satisfy Poincaré duality. In terms of the cohomology of $\mathcal{GH}^{2n+1}$, duality is given by the pairing

$$ \begin{equation} \langle[\theta],[\eta]\rangle=B(\theta,\eta),\quad\text{where}\ \ \theta\wedge\eta=B(\theta,\eta)\omega_0\wedge\omega_{-1}\wedge \omega_{1}\wedge \cdots\wedge\omega_{-n} \wedge \omega_n. \end{equation} \tag{45} $$

The dual of the bigraded component $H^{p,q}(\mathcal{GH}^{2n+1})$ is $H^{2n+1-p,-q}(\mathcal{GH}^{2n+1})$.

The Betti numbers of the manifolds $M^{2n+1}_H$ were calculated in [18]. The problem of the calculation of the cohomology $H^*(\mathcal{GH}^{2n+1})$ still attracts attention of researchers and new papers with proofs appear, using methods from quite different areas. One of the recent papers was [19]. The problem of the calculation of the integer cohomology of the lattice $\Gamma^{2n+1}_H$, that is, the integer cohomology of $M^{2n+1}_H={\mathcal H}^{2n+1} /\Gamma_H^{2n+1}$, was solved in [24]. We note that the integer homology of the space $\mathcal{H}^{2n+1}/\Gamma$, where $\Gamma\subset\mathcal{H}^{2n+1}$ is a lattice, in general, contains torsion. For $n=1$, the lattices $\Gamma$ are classified by torsion in the one-dimensional homology group. For $n>1$ there are many inequivalent lattices that are not classified by torsion in one-dimensional homology.

It is sufficient to calculate the Betti numbers $b_k=\dim H^k(\mathcal{GH}^{2n+1})$ for $p=0,\dots,n$, since $b_p=b_{2n+1-p}$ by Poincaré duality.

Theorem 16 (see [18]). (i) For $p\leqslant n$

$$ \begin{equation*} b_p=\begin{pmatrix} 2n \\ p\end{pmatrix}- \begin{pmatrix} 2n \\ p-2\end{pmatrix}. \end{equation*} \notag $$

(ii) The space of exact $p$-forms is generated by the forms $\omega_{i_1}\wedge \cdots \wedge \omega_{i_p}$ such that $i_1<\cdots <i_p$, $i_j\ne 0$ for all $j$.

(iii) The space of closed $p$-forms is generated by the forms $d\omega_{0}\wedge\omega_{i_1}\wedge \cdots \wedge \omega_{i_{p-2}}$ such that $i_1<\cdots <i_{p-2}$, $i_j\ne 0$ for all $j$.

The multiplicative structure of the cohomology $H^*(M_H^{2n+1},R)$, where $R$ is a commutative ring with unity, is quite complicated even in the case of real coefficients $R=\mathbb{R}$.

For the bundle (44) consider the Gysin exact sequence

$$ \begin{equation} \begin{aligned} \, \notag \cdots \to H^{k-2}(T^{2n},R)&\xrightarrow{\Omega} H^{k}(T^{2n},R) \xrightarrow{\pi^*} H^{k}(M^{2n+1}_H,R) \\ &\xrightarrow{\pi_*}H^{k-1}(T^{2n},R)\xrightarrow{\Omega} H^{k+1}(T^{2n},R) \to\cdots\,, \end{aligned} \end{equation} \tag{46} $$
where $\Omega$ denotes multiplication by $\Omega=d\omega_0$. It splits into short exact sequences
$$ \begin{equation} 0\to \operatorname{coker}\Omega \xrightarrow{\pi^*} H^{k}(M^{2n+1}_H,R) \xrightarrow{\pi_*} \ker \Omega \to 0. \end{equation} \tag{47} $$
Note that the cohomology groups under consideration are modules over the algebra $H^*(T^{2n},R)$, and both (46) and (47) are exact sequences of $H^*(T^{2n},R)$-modules. This gives us some information about the product in the ring $H^{*}(M^{2n+1},R)$.

To see how $\ker\Omega$ and $\operatorname{coker}\Omega$ are related to the spectral sequence of the bundle (44) we use the identification

$$ \begin{equation*} \alpha\colon E_2^{p,0} \to E_2^{p,1} ,\qquad \alpha\colon \theta\mapsto \omega_0\wedge\theta. \end{equation*} \notag $$
The composition $d_2\circ \alpha^{-1}\colon H^p(T^{2n},R)\to H^{p+2}(T^{2n},R)$ is given by $\theta\mapsto \Omega\wedge \theta$. Hence $E_\infty^{*,0}=E_3^{*,0}$ is the quotient of $H^*(T^{2n})$ by the ideal generated by $\Omega$. Furthermore, $\alpha^{-1}E_\infty^{*,1}=\alpha^{-1}E_3^{*,1}$ is the annulator of $\Omega$, that is, it consists of those classes $\theta\in H^*(T^{2n})$ for which $\Omega\wedge\theta=0$.

In the case $R=\mathbb{R}$ there is a beautiful theory that describes the groups $\ker\Omega$ and $\operatorname{coker}\Omega$ in terms of representation theory of the algebra $\mathfrak{sl}_2$ on the exterior algebra of a symplectic space. This approach was realised in [22] and [23]. We present the geometry of the exterior algebra of a symplectic vector space following the first section of [45]. See also the book [46], where the reader can find the proofs we leave out. Of course, we are talking about classical results in the theory of complex manifolds (see [72]). We also give references to the later works [45] and [46], where the reader will find all the necessary formulae.

Then we return to the discussion of the de Rham cohomology ring of $H^*(M^{2n+1}_H)$.

8.2.1. A representation of $\mathfrak{sl}_2$ on $\Lambda(\omega_{\pm1},\dots,\omega_{\pm n})$

The real cohomology algebra of the torus $T^{2n}$ is isomorphic to the exterior algebra $\Lambda(\omega_{\pm1},\dots,\omega_{\pm n})$. Let $V$ be a real $2n$-dimensional space with basis $e_{\pm 1},\dots,e_{\pm n}$. The elements of the dual basis of $V^*$ are denoted by $\omega_{\pm 1},\dots,\omega_{\pm n}$. The $2$-form $\Omega=\displaystyle\sum_{k=1}^n \omega_{-k}\wedge \omega_k$ defines a symplectic structure on $V$.

We define the operator $E^-\colon\Lambda^k V^*\to \Lambda^{k+2}V^*$ by the formula $E^-(\theta)=\Omega\wedge\theta$. Using the bivector $X_\Omega=\displaystyle\sum_{k=1}^n e_{-k}\wedge e_k\in \Lambda^2 V$ we define the operator $E^+\colon\Lambda^k V^* \to \Lambda^{k-2}V^*$ by the formula , where is the interior product of the $k$-form $\theta$ and the bivector $X_\Omega$. Finally, we consider the operator $H=[E^+,E^-]$.

Statement 6. We have

$$ \begin{equation} [H,E^+]=2E^+ \quad\text{and}\quad [H,E^-]=-2E^-. \end{equation} \tag{48} $$

Relations (48), together with the equality $H=[E^+,E^-]$, show that the choice of a symplectic structure $\Omega$ on $V$ defines a representation of the algebra $\mathfrak{sl}_2$ on the exterior algebra $\Lambda V^*$. Indeed, we have the standard basis $H$, $E^+$, $E^-$ in the Lie algebra $\mathfrak{sl}_2$ with commutation relations

$$ \begin{equation*} [E^+,E^-]=H,\qquad [H,E^+]= 2E^+\quad\text{and}\quad [H,E^-]=-2E^-. \end{equation*} \notag $$

All finite-dimensional $\mathfrak{sl}_2$-representations decompose into a direct sum of irreducible ones. There is one irreducible $\mathfrak{sl}_2$-representation in each dimension. These representations are easy to describe explicitly; see, for example, [73] (in that book the choice of basis vectors is different from ours). Let $W_m$ be a vector space with basis $v_0,v_1,\dots,v_m$. Put

$$ \begin{equation*} \begin{alignedat}{3} H(v_k) &= (m-2k) v_k,&&&& \\ E^-(v_k) &=v_{k+1}&\quad\text{for}\ \ k&< m&\quad\text{and}\quad E^-(v_m)&=0, \\ E^+(v_k) &=k(m-k+1) v_{k-1}&\quad\text{for}\ \ k&>0 &\quad\text{and}\quad E^+(v_0)&=0. \end{alignedat} \end{equation*} \notag $$
It is easy to see that these formulae define an $(m+1)$-dimensional irreducible representation of $\mathfrak{sl}_2$ in $W_m$.

Therefore, the exterior algebra $\Lambda V^*$ decomposes into a direct sum of irreducible $\mathfrak{sl}_2$-representations. This decomposition can be described using the following facts about the action of the operators $E^\pm$ and $H$ on $\Lambda V^*$.

Statement 7. (i) Let $\theta \in \Lambda^s V^*$. Them $H \theta=(n-s)\theta$.

(ii) The homomorphism $E^+\colon \Lambda^s (V^*)\to \Lambda^{s-2}(V^*)$ is injective for $s\geqslant n+1$.

(iii) The homomorphim $E^-\colon \Lambda^s (V^*)\to \Lambda^{s+2}(V^*)$ is injective for $s\leqslant n-1$.

An exterior form $\theta \in \Lambda^s V^*$, $s\leqslant n$, is called primitive if $E^+(\theta)= 0$.

The following Hodge decomposition (also known as the Hodge–Lepage decomposition) takes place.

Theorem 17. For any $\theta \in \Lambda^s(V^*)$

$$ \begin{equation*} \theta=\theta_0+E^-(\theta_1)+(E^-)^2(\theta_2)+\cdots= \theta_0+\Omega\wedge\theta_1+\Omega\wedge\Omega\wedge\theta_2+\cdots, \end{equation*} \notag $$
where $\theta_j\in \Lambda^{s-2j}(V^*)$ are uniquely defined primitive forms.

Statement 8. The maps

$$ \begin{equation*} (E^+)^s\colon\Lambda^{n+s} V^* \to \Lambda^{n-s} V^*\quad\text{and}\quad (E^-)^s\colon\Lambda^{n-s}V^*\to\Lambda^{n+s} V^* \end{equation*} \notag $$
are isomorphisms, and for any primitive form $\theta\in \Lambda^{n-s}$ the equality
$$ \begin{equation*} (E^+)^s\circ(E^-)^s (\theta)=(s!)^2 \theta \end{equation*} \notag $$
holds.

Statement 8 implies that $\Lambda^s(V^*)\subset \operatorname{im}\Omega$ for $s\geqslant n+1$, that is, non-zero elements of $\operatorname{coker}\Omega$ have grading at most $n$. Statement 7 (iii) implies that non-zero elements of $\ker\Omega$ have grading at least $n$.

Now we return to the discussion of the spectral sequence of the bundle $\pi\colon M^{2n+1}_H\to T^{2n}$. From the description of $\ker \Omega$ and $\operatorname{coker} \Omega$ just given it follows that the term $E_\infty^{*,*}$ of this spectral sequence has the form

$$ \begin{equation*} \begin{array}{c|ccccccc} 1 & 0 & \ldots & 0 & E_{\infty}^{n,1} & E_{\infty}^{n+1,0} & \ldots & E_{\infty}^{2n,1} \\ 0 & E_{\infty}^{0,0} & \ldots & E_{\infty}^{n-1,0} & E_{\infty}^{n,0} & 0 & \ldots & 0\\ \hline & 0 & \ldots & n-1 & n & n+1& \ldots & 2n \end{array} \end{equation*} \notag $$

Theorem 18. (a) The group

$$ \begin{equation*} A=\bigoplus\limits_{j=0}^n H^j (M^{2n+1}_H) \end{equation*} \notag $$
is a subring of $H^*(M_H^{2n+1})$ isomorphic to the quotient ring of the exterior algebra $\Lambda^{*}(\omega_{\pm 1},\dots,\omega_{\pm n})=H^*(T^{2n})$ by the ideal generated by $\Omega$. In other words, $A=\operatorname{coker}\Omega$ as rings and as $H^*(T^{2n})$-modules.

(b) The group

$$ \begin{equation*} B=\bigoplus\limits_{j=n+1}^{2n+1} H^j(M^{2n+1}_H) \end{equation*} \notag $$
consists precisely of those classes $\omega_0 \Lambda^{*}(\omega_{\pm 1},\dots,\omega_{\pm n})$ that lie in the kernel of multiplication by $\Omega$. In other words, $B=\ker\Omega$ as groups and as $H^*(T^{2n})$-modules. The product of two elements of $B$ is zero by dimensional considerations.

(c) The product $ab$, where $a\in A$ and $b\in B$, belongs to $B$ and defines a representation of the ring $A$ in the module $B$. Hence $H^*(M_H^{2n+1})$ is obtained by extending the ring $A$ by the action of $A$ on $B$.

(d) The splitting $H^*(M_H^{2n+1})=A\oplus B$ is a splitting of $H^*(T^{2n})$-modules.

Corollary 8. (i) As an $H^*(T^{2n})$-module, the graded group $H^*(M_H^{2n+1})$ is generated by $1\in H^0(M_H^{2n+1})$ and the generators of $H^{n+1}(M_H^{2n+1})$.

(ii) The cohomology ring $H^*(M_H^{2n+1})$ is multiplicatively generated by the generators $\omega_{\pm 1},\dots,\omega_{\pm n}$ of $H^1(M_H^{2n+1})$ and the generators of $H^{n+1}(M_H^{2n+1})$.

Theorem 18 implies, in particular, that the short exact sequence (47)

$$ \begin{equation} 0\to \operatorname{coker}\Omega \xrightarrow{\pi^*} H^{k}(M^{2n+1}_H,\mathbb{R}) \xrightarrow{\pi_*} \ker \Omega \to 0 \end{equation} \tag{49} $$
splits as a sequence of $H^*(T^{2n},\mathbb{R})$-modules. The following important fact holds.

Theorem 19 ([74]). Let $S^m\to X\xrightarrow{\pi} T$ be an orientable sphere bundle over a torus, and let $\Omega\in H^{m+1}(T,\mathbb{Z}) $ be its characteristic class. Then for any $k$ the exact sequence

$$ \begin{equation*} 0\to \operatorname{coker}\Omega \xrightarrow{\pi^*} H^{k}(X,\mathbb{Z}) \xrightarrow{\pi_*} \ker \Omega \to 0 \end{equation*} \notag $$
splits as a sequence of $H^*(T,\mathbb{Z})$-modules.

For integer coefficients this statement is quite non-trivial; it requires an analysis of the cohomological obstruction to splitting of a short exact sequence and the proof that this obstruction is zero.

Now we describe the relationship between the Poincaré polynomials of the cohomology of the Lie algebras $\mathcal{GH}^{2n-1}$ in $\mathcal{GH}^{2n+1}$. Consider the Hochschild–Serre spectral sequence of the Lie subalgebra $\mathcal{GH}^{2n-1}$ in $\mathcal{GH}^{2n+1}$. The quotient $\mathcal{GH}^{2n+1}/\mathcal{GH}^{2n+1}$ is a two-dimensional commutative Lie algebra $\mathfrak{t}^2$ with generators $\omega_{\pm n}$, so the second term of this spectral sequence is

$$ \begin{equation} E_2^{p,q}=H^p(\mathfrak{t}^2,H^q(\mathcal{GH}^{2n-1})). \end{equation} \tag{50} $$
It is easy to check that the action $\mathfrak{t}^2$ on $H^q(\mathcal{GH}^{2n-1})$ is trivial, so the spectral sequence (50) has the form

The group $H^s(\mathcal{GH}^{2n-1})$ will be denoted simply by $H^s$.

Statement 9. In the Hochschild–Serre spectral sequence of the Lie subalgebra $\mathcal{GH}^{2n-1}$ in $\mathcal{GH}^{2n+1}$ all differentials vanish, except for $d_2\colon E_2^{0,n}\to E_2^{2,n-1}$ (indicated by an arrow in the diagram), which is an isomorphism.

Corollary 9. The Poincaré polynomials $P_{n}(t)$ and $P_{n-1}(t)$ of the cohomology of the Lie algebras $\mathcal{GH}^{2n+1}$ and $\mathcal{GH}^{2n-1}$ are related by the recurrence relation

$$ \begin{equation} P_{n}(t)=(1+t)^2P_{n-1}(t)-\biggl(\begin{pmatrix} 2n-2 \\ n-1\end{pmatrix}- \begin{pmatrix} 2n-2 \\ n-3\end{pmatrix}\biggr)(1+t)t^{n}. \end{equation} \tag{51} $$

Proof. Let $b_k=\dim H^k(\mathcal{GH}^{2n-1})$ and $\widehat b_k=\dim H^k(\mathcal{GH}^{2n+1})$. By Poincaré duality it suffices to compare the coefficients of $t^k$ in (51) for $0\leqslant k \leqslant n$.

For $k=0$ and $k=1$ this is quite simple: $\widehat b_0=b_0=1$ and $\widehat b_1=b_1+2=2n$.

For $2\leqslant k\leqslant n-1$ we need to check the equality $\widehat b_k=b_k+ 2 b_{k-1} + b_{k-2}$. To do this we write the Betti numbers as the differences between two binomial coefficients in accordance with Theorem 16 and use the formula ${N \choose p}+ {N\choose p-1}={N+1\choose p}$ several times. It follows that all differentials $d_2$ below the one shown in the diagram are zero. By Poincaré duality the differentials $d_2$ above the one shown are zero too.

On the other hand it is easy to verify that $\widehat b_{n}$ is not equal to $b_{n} + 2 b_{n-1} + b_{n-2}=3 b_{n-1} + b_{n-2}$. This means that the differential $d_2\colon E_2^{0,n+1}\to E_2^{2,n}$ in the spectral sequence is non-trivial. Let us show that it is in fact an isomorphism. To do this note that $\widehat b_{n}=2 b_{n-1} + b_{n-2}$. This is equivalent to the identity

$$ \begin{equation*} \begin{pmatrix} 2n \\ n-2\end{pmatrix}+ \begin{pmatrix} 2n-1 \\ n-1\end{pmatrix}+ \begin{pmatrix} 2n-2 \\ n-1 \end{pmatrix}= \begin{pmatrix} 2n \\ n\end{pmatrix}+ \begin{pmatrix}2n-1 \\ n-3\end{pmatrix}+ \begin{pmatrix} 2n-2 \\ n-3 \end{pmatrix}. \end{equation*} \notag $$
It follows that the groups $E_3^{0,n+2}$ and $ E_3^{2,n+1}$ are zero, that is, $d_2\colon E_2^{0,n+2}\to E_2^{2,n+1}$ is an isomorphism.

The Poincaré polynomial of $\bigoplus\limits_{p+q=n} E_2^{p,q}$ is equal to $(1+t)^2P_{n-1}(t)$. In going over to $E_3$ we must take into account the single non-trivial differential, that is, we must subtract $b_{n-1}(1+t)t^{n}$ from the resulting expression. Since $E_3= E_\infty$, this completes the proof of formula (51). $\Box$

8.3. The $\operatorname{Bss}$ for the Lie algebras $\mathcal{GH}^{2n+1}$

We formulate and prove several general facts about the $\operatorname{Bss}$ for $\mathcal{GH}^{2n+1}$.

The universal enveloping algebra $U\mathcal{GH}^{2n+1}$ has a basis consisting of elements $e_-^ae_+^b e_0^c=e_{-1}^{a_1} \cdots e_{-n}^{a_n} e_{1}^{b_1}\cdots e_{n}^{b_n}e_0^c$, where we use the notation $a=(a_1,\dots,a_{n})$ and $b=(b_1,\dots,b_{n})$. Elements of the dual basis of $(U\mathcal{GH}^{2n+1})^*$ are denoted by $(e_-^ae_+^b e_0^c)^*$. Consider the Buchstaber filtration starting with $N_0=\mathbb{R}$.

Lemma 15. Let

$$ \begin{equation*} p(a,b,c)=p(a_1,\dots,a_{n},b_1,\dots,b_n,c)= 2c+\sum_{j=1}^{n} a_j+\sum_{j=1}^{n} b_j. \end{equation*} \notag $$

(i) An element $(e_-^ae_+^b e_0^c)^*$ with $p=p(a,b,c)$ belongs to $N_p$ and does not belong to $N_{p-1}$.

(ii) Furthermore, $N_{p }=N_{p-1} \oplus \mathbb{R}\langle (e_-^ae_+^b e_0^c)^* \colon p(a,b,c)= p\rangle$.

Proof. This is similar to the proof of Theorem 7. $\Box$

Corollary 10. The generating series of the sequence $k_p=\dim N_p/N_{p-1}$ is

$$ \begin{equation*} \sum_{p\geqslant 0}k_p x^p=\frac{1}{(1-x)^{2n}(1-x^2)}\,. \end{equation*} \notag $$

The Chevalley–Eilenberg differential on the complex

$$ \begin{equation*} (U\mathcal{GH}^{2n+1})^*\otimes \wedge(\omega_{\pm 1},\dots,\omega_{\pm n},\omega_0) \end{equation*} \notag $$
increases the first grading by 1, preserves the second grading, and can decrease the filtration by at most $2$. This means precisely that the differentials $d_k$ in the corresponding $\operatorname{Bss}$ are trivial for $k\geqslant 3$; in other words, $E_3^{*,*,*}=E_\infty^{*,*,*}$ (see Theorem 5).

Recall that in § 3.3 we defined an increasing filtration $\Phi^p$ on the cohomology groups $H^k(\mathcal{GH}^{2n+1})$ by using the differentials of the trigraded Buchstaber spectral sequence. Since the differentials $d_k$ are trivial for $k\geqslant 3$, the filtration stabilises at $\Phi^2$, namely, $\Phi^2=\Phi^k$ for all $k\geqslant 2$.

Theorem 20. (i) The elements of $H^k(\mathcal{GH}^{2n+1})$ for $k\ne n+1$ belong to $\Phi^1$.

(ii) The elements of $H^{n+1}(\mathcal{GH}^{2n+1})$ belong to $\Phi^2$, but do not belong to $\Phi^1$.

Proof. It follows from Lemma 4 that $H^1(\mathcal{GH}^{2n+1})\subset \Phi^1$.

We prove part (i) using induction on $n\geqslant 1$. The basis ($n=1$) follows from the explicit description of the basis cocycles of the cohomology of $\mathcal{GH}^3$:

$$ \begin{equation*} \begin{alignedat}{2} k&=0:&&\quad 1; \\ k&=1:&&\quad \omega_{-1},\ \ \omega_1; \\ k&=2:&&\quad \omega_0 \wedge\omega_{-1},\ \ \omega_0\wedge\omega_1; \\ k&=3:&&\quad \omega_0\wedge \omega_{-1}\wedge \omega_1. \end{alignedat} \end{equation*} \notag $$
The basic cocycles of the cohomology of $\mathcal{GH}^5$ are presented in § 8.4.

Next, we consider the Hochschild–Serre spectral sequence (50). By Statement 9 the term $E_\infty$ has the form

$$ \begin{equation*} \begin{array}{c|ccc} 2n-1 & H^{2n-1} & \omega_{-n}H^{2n-1}\oplus \omega_{n}H^{2n-1}& \omega_{-n}\wedge\omega_{n}H^{2n-1}\\ \vdots & \vdots & \vdots & \vdots \\ n & 0 & \omega_{-n}H^{n}\oplus \omega_{n}H^{n}& \omega_{-n}\wedge\omega_{n}H^{n} \vphantom{\biggl\}}\\ n-1 & H^{n-1} & \omega_{-n}H^{n-1}\oplus \omega_{n}H^{n-1}& 0 \vphantom{\biggl\}}\\ \vdots & \vdots & \vdots & \vdots \\ 0 & H^{0} & \omega_{-n}H^{0}\oplus \omega_{n}H^{0}\vphantom{\biggl\}}& \omega_{-n}\wedge\omega_{n}H^{0}\\ \hline & 0 & 1 & 2 \end{array} \end{equation*} \notag $$

For $k\geqslant n+2$ or $k\leqslant n$ there is a three-term filtration in the group $H^{k}(\mathcal{GH}^{2n+1})$, with the associated group given by

$$ \begin{equation*} H^k(\mathcal{GH}^{2n-1})\oplus \omega_{-n}H^{k-1}(\mathcal{GH}^{2n-1}) \oplus \omega_{n}H^{k-1}(\mathcal{GH}^{2n-1})\oplus \omega_{-n}\wedge\omega_n H^{k-2}(\mathcal{GH}^{2n-1}). \end{equation*} \notag $$
The proof of part (i) follows by induction.

To prove part (ii) observe that the image of $d_1\colon N_1/N_0\otimes H^{n}(\mathcal{GH}^{2n+1}) \to H^{n+1}(\mathcal{GH}^{2n+1})$ is zero, since any cochain representing an element of dimension $n+1$ must contain $\omega_{0}$; this follows from the non-degeneracy of the pairing (45). Hence there are no non-trivial elements of filtration $\Phi^1$ in $H^{n+1} (\mathcal{GH}^{2n+1})$. $\Box$

In [14] elements of the group $H^{n+1}(\mathcal{GH}^{2n+1})$ representable by matric Massey products were specified.

Statement 10 ([14], § 2.5). The cochains $\omega_0\wedge \omega_{\varepsilon_1\cdot 1}\wedge \omega_{\varepsilon_2\cdot 2}\wedge\cdots\wedge\omega_{\varepsilon_n\cdot n}$ such that $\varepsilon_j=\pm 1$ represent linearly independent classes in $H^{n+1}(\mathcal{GH}^{2n+1})$. These classes do not decompose into linear combinations of non-trivial products of classes of lower dimension, but are representatives of non-trivial triple matric Massey products. Namely, the class of the cocycle $\omega_0\wedge \omega_{\varepsilon_1\cdot 1}\wedge \omega_{\varepsilon_2\cdot 2}\wedge\cdots\wedge\omega_{\varepsilon_n\cdot n}$ belongs to

$$ \begin{equation*} \Biggl\langle(\varepsilon_1\omega_{-\varepsilon_1\cdot 1},\varepsilon_2 \omega_{-\varepsilon_2\cdot 2},\dots,\varepsilon_n \omega_{-\varepsilon_n\cdot n}),\begin{pmatrix} \omega_{\varepsilon_1\cdot 1} \\ \omega_{\varepsilon_2\cdot 2}\\ \vdots \\ \omega_{\varepsilon_n\cdot n} \end{pmatrix},\omega_{\varepsilon_1\cdot 1} \wedge \omega_{\varepsilon_2\cdot 2}\wedge\cdots \wedge \omega_{\varepsilon_n\cdot n}\Biggr\rangle. \end{equation*} \notag $$

This statement follows by setting $U=-\omega_0$ and $V=0$ in the notation of § 6.1.

It is easy to indicate which elements are mapped to the elements described in Statement 10 by the differential $d_2$.

Statement 11. The cohomology class in $H^{n+1}(\mathcal{GH}^{2n+1})$ represented by the cocycle $\omega_0\wedge \omega_{\varepsilon_1\cdot 1}\wedge \omega_{\varepsilon_2\cdot 2}\wedge\cdots \wedge \omega_{\varepsilon_n\cdot n}$ is equal to

$$ \begin{equation*} d_2\biggl(\biggl(e_0^*+\sum_k\delta_{\varepsilon_k}^1(e_{-k}e_k)^*\biggr) \otimes\omega_{\varepsilon_1\cdot 1}\wedge \omega_{\varepsilon_2\cdot 2} \wedge\cdots \wedge\omega_{\varepsilon_n\cdot n}\biggr), \end{equation*} \notag $$
where $\delta_i^j$ is the Kronecker delta.

The proof follows from identities in the complex $(U\mathcal{GH}^{2n+1})^*\otimes \wedge (\omega_{\pm 1},\dots,\omega_{\pm n},\omega_0)$:

$$ \begin{equation*} d( (e_0)^*\otimes \omega)=1\otimes \omega_0\wedge \omega - \sum_{k=1}^n (e_k)^*\otimes \omega_{-k}\wedge\omega \end{equation*} \notag $$
and
$$ \begin{equation*} d( (e_{-k}e_k)^*\otimes \omega)=(e_k)^*\otimes \omega_{-k}\wedge \omega + (e_{-k})^*\otimes \omega_{k}\wedge \omega. \end{equation*} \notag $$

Note that the number of classes described in Statements 10 and 11 is $2^n$, which is less than the corresponding Betti number $b_{n+1}$. This number, by virtue of Poincaré duality and Theorem 16, is equal to

$$ \begin{equation*} b_{n+1}=b_n={2n\choose n}-{2n\choose n-2}. \end{equation*} \notag $$
In other words, the group $H^{n+1}(\mathcal{GH}^{2n+1})$ contains elements not described in Statements 10 and 11. According to Theorem 20, all non-zero elements of $H^{n+1}(\mathcal{GH}^{2n+1})$ belong to $\Phi^2$, but do not belong to $\Phi^1$. As we show in § 9 (see Theorem 21), these elements are represented by triple matric Massey products of a special form. In §§ 8.4 and 8.5 below we represent the remaining elements of the group $H^{n+1} (\mathcal{GH}^{2n+1})$ for $n=2$ and $n=3$ by triple matric Massey products.

8.4. The Lie algebra $\mathcal{GH}^{5}$

By Nomizu’s theorem, the real cohomology ring of $M_H^{2n+1}$ is isomorphic to the cohomology ring $H^*(\mathcal{GH}^{2n+1})$. In this section we consider the case of the algebra $\mathcal{GH}^{5}$. As before, we denote the element dual to $e_k$ by $\omega_k$. Recall that the differential $d$ in the Chevalley–Eilenberg complex $\wedge(\mathcal{GH}^5)^*$ is defined by the relations

$$ \begin{equation*} d\omega_{\pm 1}=d\omega_{\pm 2}=0, \qquad d\omega_0=\omega_{-1}\wedge\omega_{1}+\omega_{-2}\wedge\omega_2. \end{equation*} \notag $$
It follows easily that the real cohomology $H^k(\mathcal{GH}^{5})$ has the following bases:
$$ \begin{equation*} \begin{alignedat}{2} k&=0:&&\quad 1; \\ k&=1:&&\quad \omega_{-1},\ \ \omega_{-2},\ \ \omega_2,\ \ \omega_1; \\ k&=2:&&\quad \omega_{-1}\wedge \omega_{-2},\ \ \omega_{-1}\wedge\omega_2,\ \ \omega_{-1}\wedge\omega_1=-\omega_{-2}\wedge\omega_2,\ \ \omega_{-2}\wedge\omega_1,\ \ \omega_1\wedge\omega_2; \\ k&=3:&&\quad \omega_0 \wedge\omega_{-1}\wedge\omega_{-2},\ \ \omega_0\wedge\omega_{-1}\wedge\omega_2,\ \ \omega_0\wedge\omega_{-2}\wedge\omega_1,\ \ \omega_0\wedge\omega_1\wedge\omega_2, \\ &&&\quad\omega_0\wedge(\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2); \\ k&=4:&&\quad \omega_0\wedge \omega_{-2}\wedge \omega_1\wedge \omega_2,\ \ \omega_0 \wedge\omega_{-1}\wedge \omega_1\wedge \omega_{2},\ \ \omega_0\wedge \omega_{-1}\wedge \omega_{-2}\wedge \omega_1, \\ &&&\quad\omega_{0} \wedge \omega_{-1}\wedge \omega_{-2}\wedge\omega_2; \\ k&=5:&&\quad \omega_0\wedge \omega_{-1}\wedge \omega_{-2}\wedge \omega_1\wedge \omega_2. \end{alignedat} \end{equation*} \notag $$

A direct check shows that all elements of dimensions $1$, $2$, $4$, and $5$ have filtration degree $1$.

The elements of $H^3(\mathcal{GH}^{5})$ represented by the cocycles $\omega_0\wedge\omega_{-1}\wedge\omega_{-2}$, $\omega_0\wedge\omega_{-1}\wedge\omega_2$, $\omega_0\wedge\omega_{-2}\wedge\omega_1$, $\omega_0\wedge\omega_1\wedge\omega_2$ belong to the filtration $\Phi^2$. As shown in Statements 10 and 11, these cohomology classes are represented by non-trivial triple Massey products and also lie in the image of the differential $d_2$ of the Buchstaber spectral sequence.

For the remaining class $\omega_0\wedge(\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2)$ a similar statement holds:

Statement 12. (i) The cocycle $\omega_0\wedge(\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2)$ is a representative of the triple matric Massey product

$$ \begin{equation*} \biggl\langle \begin{pmatrix}\omega_{-1} & \omega_{-2}\end{pmatrix}, \begin{pmatrix} (1/2)\omega_1 \\ (1/2)\omega_2 \end{pmatrix}, \omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2\biggr\rangle. \end{equation*} \notag $$

(ii) The class represented by the form $\omega_0\wedge(\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2)$ is equal to

$$ \begin{equation*} d_2\bigl((e_0)^*\otimes (\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2) -(e_2e_1)^*\otimes \omega_{-1}\wedge\omega_{-2}\bigr). \end{equation*} \notag $$

Proof. To prove (i) we put
$$ \begin{equation*} A=\begin{pmatrix}\omega_{-1} & \omega_{-2}\end{pmatrix}, \qquad B=\begin{pmatrix} (1/2)\omega_1 \\ (1/2)\omega_2 \end{pmatrix},\qquad C=\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2. \end{equation*} \notag $$

Then $\bar{A}B=-(1/2)(\omega_{-1}\wedge\omega_1+ \omega_{-2}\wedge\omega_2)$ is an exact form. Namely, for $U=-(1/2)\omega_0$ we have $\bar{A}B=dU$.

Further, the matrix $\overline{B} C=\begin{pmatrix} \hphantom{-}(1/2)\omega_1\wedge\omega_{-2}\wedge\omega_2 \\ -(1/2)\omega_2\wedge\omega_{-1}\wedge\omega_1 \end{pmatrix}$ consists of exact forms. Namely, we have $dV=\overline{B}C$ for $V=\begin{pmatrix} -(1/2)\omega_1\wedge \omega_0 \\ \hphantom{-}(1/2)\omega_2\wedge \omega_0 \end{pmatrix}$.

The triple Massey product $\langle A,B,C\rangle$ contains the element

$$ \begin{equation*} \begin{aligned} \, \overline{U} C + \bar{A} {V}&= \frac{1}{2}\omega_0\wedge(\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2) -\frac{1}{2}\omega_{-1}\wedge(-\omega_1\wedge\omega_0) \\ &\qquad-\frac{1}{2}\omega_{-2}\wedge(\omega_2\wedge\omega_0)= \omega_0\wedge(\omega_{-1}\wedge\omega_1-\omega_{-2}\wedge\omega_2), \end{aligned} \end{equation*} \notag $$
which proves (i).

To prove part (ii) we calculate

$$ \begin{equation*} \begin{aligned} \, &d_{\rm CE}\bigl((e_0)^*\otimes (\omega_{-1}\wedge\omega_1- \omega_{-2}\wedge\omega_2)\bigr)=\omega_0\wedge (\omega_{-1}\wedge\omega_1- \omega_{-2}\wedge\omega_2) \\ &\qquad +(e_1)^*\otimes \omega_{-1}\wedge\omega_{-2}\wedge\omega_2 - (e_2)^*\otimes \omega_{-2}\wedge\omega_{-1}\wedge\omega_1, \\ &d_{\rm CE}\bigl((e_2e_1)^*\otimes \omega_{-1}\wedge\omega_{-2}\bigr)= (e_2)^*\otimes \omega_1\wedge\omega_{-1}\wedge\omega_{-2} \\ &\qquad+(e_1)^*\otimes \omega_2\wedge\omega_{-1}\wedge\omega_{-2}. \end{aligned} \end{equation*} \notag $$
Subtracting the second relation from the first we obtain the equality proving (ii). $\Box$

8.5. The Lie algebra $\mathcal{GH}^{7}$

Recall that the differential $d$ in the Chevalley–Eilenberg complex $\wedge (\mathcal{GH}^{7})^*$ is defined by the relations

$$ \begin{equation*} d\omega_{\pm 1}=d\omega_{\pm 2}=d\omega_{\pm 3}=0\quad\text{and}\quad d\omega_0=\omega_{-1}\wedge\omega_{1}+ \omega_{-2}\wedge\omega_2+\omega_{-3}\wedge\omega_3. \end{equation*} \notag $$

From here we could easily obtain explicitly basis elements of the cohomology groups $H^k(\mathcal{GH}^{7})$. Instead, we make the following general observation.

As we already know, for $k=1,2,3,5,6,7$ all elements of the groups $H^k(\mathcal{GH}^{7})$ belong to the filtration $\Phi^1$, in particular, they lie in the image of the differential $d_1$ of the Buchstaber spectral sequence.

All elements of the group $H^4(\mathcal{GH}^{7})$ belong to the filtration $\Phi^2$, in particular, they lie in the image of the differential $d_2$ of the Buchstaber spectral sequence.

Statements 10 and 11 imply that the eight independent elements of $H^4(\mathcal{GH}^{7})$ represented by cocycles of the form $e_0\wedge \omega_{\pm1}\wedge\omega_{\pm 2}\wedge\omega_{\pm 3}$ are representable as non-trivial triple Massey products. It has also been shown explicitly that each of these elements belongs to the image of $d_2$.

On the other hand $b_4=b_3={6\choose 4}-{6\choose 2}=14$, so $H^4(\mathcal{GH}^{7})$ contains six independent classes more. They are easy to specify explicitly (below we write $(i_1,i_2,\ldots)$ instead of $\omega_{i_1}\wedge\omega_{i_2}\wedge\cdots$):

$$ \begin{equation} \begin{aligned} \, &(0,-1,1,2)+(0,-3,2,3), \\ &(0,-2,1,2)-(0,-3,1,3), \\ &(0,-2,2,3)-(0,-1,1,3), \\ &(0,-2,-3,3)+(0,-1,-2,1), \\ &(0,-1,-3,3)-(0,-1,-2,2), \\ &(0,-1,-3,1)-(0,-2,-3,2). \end{aligned} \end{equation} \tag{52} $$

Each of these elements belongs to the image of $d_2$ and is also represented by a triple Massey product. We demonstrate this only for the element $(0,-1,1,2)+(0,-3,2,3)$; the remaining five elements are treated similarly. First we show that this element belongs to the triple Massey product

$$ \begin{equation*} \left\langle\begin{pmatrix} \omega_{-1} & \omega_{-2} & \omega_{-3} \end{pmatrix},\begin{pmatrix} (1/2)\omega_1 \\ (1/2)\omega_2 \\ (1/2)\omega_3 \end{pmatrix},\,\omega_{-1}\wedge\omega_1\wedge\omega_2+ \omega_{-3}\wedge\omega_2\wedge\omega_3\right\rangle. \end{equation*} \notag $$
We put
$$ \begin{equation*} A=\begin{pmatrix} \omega_{-1} & \omega_{-2} & \omega_{-3} \end{pmatrix}, \qquad B=\begin{pmatrix} (1/2)\omega_1 \\ (1/2)\omega_2 \\ (1/2)\omega_3 \end{pmatrix}, \end{equation*} \notag $$
and
$$ \begin{equation*} C=\omega_{-1}\wedge\omega_1\wedge\omega_2+ \omega_{-3}\wedge\omega_2\wedge\omega_3. \end{equation*} \notag $$
Then $\bar{A}B=-(1/2)(\omega_{-1}\wedge\omega_1+\omega_{-2}\wedge\omega_2+ \omega_{-3}\wedge\omega_3)$ is an exact form. Namely, we have $\bar{A}B=d U$ for $U=-(1/2)\omega_0$.

Further, the matrix

$$ \begin{equation*} \overline {B} C=\begin{pmatrix} -(1/2)\omega_1\wedge\omega_{-3}\wedge\omega_2\wedge\omega_3 \\ 0 \\ -(1/2)\omega_3\wedge\omega_{-1}\wedge\omega_{1}\wedge\omega_2 \end{pmatrix} \end{equation*} \notag $$
consists of exact forms. Namely, we have $dV=\overline{B}C$ for
$$ \begin{equation*} V=\begin{pmatrix} (1/2)\omega_0\wedge\omega_1\wedge\omega_2 \\ 0 \\ -(1/2)\omega_0\wedge\omega_3\wedge \omega_2 \\ \end{pmatrix}. \end{equation*} \notag $$

The triple Massey product $\langle A,B,C\rangle$ contains the element

$$ \begin{equation*} \begin{aligned} \, \overline{U} C+\bar{A}{V}&= \frac{1}{2}\omega_0\wedge(\omega_{-1}\wedge\omega_1\wedge\omega_2+ \omega_{-3}\wedge\omega_2\wedge\omega_3) \\ &\qquad-\frac{1}{2}\omega_{-1}\wedge\omega_0\wedge\omega_1\wedge\omega_2+ \frac{1}{2}\omega_{-3}\wedge\omega_0\wedge\omega_3\wedge\omega_2 \\ &=\omega_0\wedge(\omega_{-1}\wedge\omega_1\wedge\omega_2- \omega_{-3}\wedge\omega_{2}\wedge\omega_3), \end{aligned} \end{equation*} \notag $$
as required.

Second, we show that the element $(0,-1,1,2)+(0,-3,2,3)$ is equal to

$$ \begin{equation*} \begin{aligned} \, &d_2\bigl((e_0)^*\otimes\bigl((-1,1,2)+(-3,2,3)\bigr)+ (e_1e_3)^*\otimes(-3,-1,2) \\ &\qquad+(e_{-2}e_2)^*\otimes \bigl((-3,2,3)+(-1,1,2)\bigr)\bigr). \end{aligned} \end{equation*} \notag $$
To do this we calculate the Chevalley-Eilenberg differential of each of these three terms:
$$ \begin{equation*} \begin{aligned} \, &d_{\rm CE}\bigl((e_0)^*\otimes\bigl((-1,1,2)+(-3,2,3)\bigr)\bigr) = (0,-1,1,2)+(0,-3,2,3) \\ &\qquad\qquad -(e_2)^*\otimes (-2,-1,1,2)-(e_3)^* \otimes (-3,-1,1,2) \\ &\qquad\qquad -(e_1)^*\otimes (-1,-3,2,3)-(e_2)^*\otimes (-2,-3,2,3); \\ &d_{\rm CE}\bigl((e_1e_3)^*\otimes (-3,-1,2)\bigr) = (e_1)^*\otimes (3,-3,-1,2) + (e_3)^*\otimes (1,-3,-1,2); \\ &d_{\rm CE}\bigl((e_{-2}e_2)^*\otimes ((-3,2,3)+(-1,1,2))\bigr) \\ &\qquad=(e_{-2})^*\otimes \bigl((-2,-3,2,3)+(-2,-1,1,2)\bigr). \end{aligned} \end{equation*} \notag $$
It remains to add these three equalities.

9. Realisation of $\operatorname{Bss}$-operations by special Massey products

9.1. Special matric Massey products

Let $A$ be a Hopf algebra over a ring $R$. The Buchstaber filtration $\{N^k\subset A_*\colon k\geqslant 0\}$ starting with $N^0=R$ is defined on the dual Hopf algebra $A_*$ (see Definition 1 and formula (9)). We denote the spectal sequence $E_r^{*,*,*}$ defined by this filtration by $\operatorname{Bss}$.

Recall that the differentials $d_r\colon E_r^{0,*,*}\to E_r^{-r,*+r-1,*}$ are trivial for $r\geqslant 1$, and therefore the images of the differentials $d_r\colon E_r^{r,*-r+1,*}\to E_r^{0,*,*}$ define a filtration $\Phi^r$ in the groups $E_1^{0,-s,t}=\operatorname{Ext}^{s,t}_A(R,R)$ (see § 3.3). Recall its simplest properties:

(i) $\Phi^0=R=E^{0,0,0}_\infty$;

(ii) the quotient $\Phi^r/\Phi^{r-1}\subset\operatorname{Ext}_A^{q,*}(R,R)/\Phi^{r-1}$ coincides with the image of

$$ \begin{equation*} d_r\colon E_r^{r,*-r+1,*}\to E_r^{0,*,*}; \end{equation*} \notag $$

(iii) $\operatorname{Ext}_A^{q,*}(R,R)/\Phi^{r-1}=E_r^{0,-q,*}$.

The following theorem gives a solution to the problem of realisation of $\operatorname{Bss}$-operations by non-trivial matric Massey products. Namely, for a Hopf algebra $A$ over a field $\Bbbk$, let $x\in \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ be a cohomology class of filtration degree exactly $k$. By Theorem 4 the class $x$ is realised by a non-trivial $\operatorname{Bss}_k$-operation. We will show that the same class $x$ is realised by a non-trivial Massey product.

Theorem 21. Let $A$ be a Hopf algebra over a field $\Bbbk$. Let $x\in \Phi^k \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$, but $x\notin \Phi^{k-1} \operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$. Then $x$ is a representative of a non-trivial matric $(k+1)$-fold Massey product of the form $\langle a_1,a_2,\dots,a_k,a_{k+1} \rangle$, where the matrices $a_1,a_2,\dots,a_k$ are formed of elements of $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$, and the matrix $a_{k+1}$ is formed of elements of $\operatorname{Ext}^{s-1,*}_A(\Bbbk,\Bbbk)$.

An important property of the filtration $\{\Phi^r\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)\colon r\geqslant 0\}$ is that it exhausts each group $\operatorname{Ext}^{s,t}_A(\Bbbk,\Bbbk)$, in other words, any element of $\operatorname{Ext}^{s,t}_A(\Bbbk,\Bbbk)$ satisfies the assumptions of Theorem 21. Indeed, $E_\infty^{*,*,*}$ is zero, except for $E_\infty^{0,0,0}=\Bbbk$, and therefore the images of the differentials $d_r$, $r\geqslant 1$, exhaust the groups $E_1^{0,-s,t}=\operatorname{Ext}^{s,t}_A(\Bbbk,\Bbbk)$.

We introduce the concepts of a special matric Massey product and a special iterated matric Massey product.

For $s\geqslant 2$, a special matric Massey product is a Massey product of the form $\langle a_1,a_2,\dots,a_k,y \rangle$, where $a_1,a_2,\dots,a_k$ are matrices of elements of $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$, and $y$ is a matrix of elements of $\operatorname{Ext}^{s-1,*}_A(\Bbbk,\Bbbk)$. For $k\geqslant 1$ a special matric Massey product consists of elements of $\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$.

For $s=2$, a special iterated Massey product is a special Massey product for which $y$ is a matrix of elements of $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$.

For $s>2$, a special iterated Massey product is a special Massey product such that $y$ is a matrix of elements of $\operatorname{Ext}^{s-1,*}_A(\Bbbk,\Bbbk)$ which are representatives of special iterated Massey products.

Note that for $s=2$ a special matric Massey product does not contain iterations.

A special iterated Massey product is called non-trivial if all Massey products included in it are non-trivial, that is, do not contain zero.

Corollary 11. Let $A$ be a Hopf algebra over a field $\Bbbk$. Then any element of the cohomology group $\operatorname{Ext}^{s,*}_A(\Bbbk,\Bbbk)$ for $s\geqslant 2$ can be expressed as a non-trivial special iterated matric Massey product of elements of $\operatorname{Ext}^{1,*}_A(\Bbbk,\Bbbk)$.

Recall that multiplication in $\operatorname{Ext}^{*,*}_{UL_1}(\mathbb{Q},\mathbb{Q})$ is trivial (see § 2), so binary Massey products do not contain any elements other than zero. Moreover, $\operatorname{Ext}^{1,*}_{UL_1}(\mathbb{Q},\mathbb{Q})$ has exactly two generators $x_{\pm}^1$. Therefore, as an application of Theorem 21 and Corollary 11, we obtain a proof of the ‘matric’ Buchstaber conjecture: any cohomology class in $\operatorname{Ext}^{s,*}_{UL_1}(\mathbb{Q},\mathbb{Q})$, where $s\geqslant 2$, can be represented as a non-trivial special iterated matric $k$-fold Massey product of elements $x_{\pm}^1$ only, where $k\geqslant 3$.

Remark 9. As will be seen from the proof of Theorem 21, the key fact is that the filtration $\{N_p\}$ exhausts $A_*$, so Theorem 21 remains true for any residually nilpotent Lie algebra without grading, in particular, for any finite-dimensional nilpotent Lie algebra. In this case the matric version of Buchstaber’s conjecture also holds: any cohomology class of such a Lie algebra $\mathfrak{g}$ can be expressed as an iterated non-trivial Massey product of classes from $H^1(\mathfrak{g})$ (here two-fold Massey products are also used if multiplication in the cohomology of $\mathfrak{g}$ is non-trivial).

9.2. Realisation of $\operatorname{Bss}_p$-operations

We fix the following notation:

$\bullet$ the complex $F^{\small\bullet}(A_*,A_*,\Bbbk)$ (see § 3.2) will be denoted by $\widetilde{F}^{\small\bullet}(A_*)$; recall that this is a contractible complex;

$\bullet$ the complex $F^{\small\bullet}(\Bbbk,A_*,\Bbbk)$ will be denoted by $F^{\small\bullet}(A_*)$;

$\bullet$ an element $a_0\otimes a_1\otimes \cdots\otimes a_s\in \widetilde{F}^s(A_*)$ will be denoted by $a_0\otimes [\kern-1.5pt[ a_1\otimes \cdots \otimes a_s ]\kern-1.5pt] $;

$\bullet$ an element $a_1\otimes \cdots\otimes a_s\in F^s(A_*)$ will be denoted by $ [\kern-1.5pt[ a_1\otimes \cdots \otimes a_s ]\kern-1.5pt] $;

$\bullet$ $F^{\small\bullet}(A_*)$ will be considered as a subcomplex of $\widetilde{F}^{\small\bullet}(A_*)$; under this inclusion $ [\kern-1.5pt[ a_1\otimes \cdots \otimes a_s ]\kern-1.5pt] \in F^s(A_*)$ is mapped to $1\otimes [\kern-1.5pt[ a_1\otimes \cdots \otimes a_s ]\kern-1.5pt] \in\widetilde{F}^s(A_*)$.

We choose a special basis $\{m_{i_p}(p)\}$ in $A_*$ compatible with the filtration $\{N_p\}$ as follows. In $N_0$ there is exactly one basis element, $m_1(0)=1\in N_0$. We choose homogeneous elements $m_{i_1}(1)$ in $N_1$, where $1\leqslant i_1\leqslant \dim N_1/N_0$, so that $\{m_1(0)\}\cup \{m_{i_1}(1)\}$ is a basis of $N_1$. Suppose that we have chosen a basis $\{m_{i_k}(k)\colon k<p\}$ in $N_{p-1}$. Then we choose homogeneous elements $m_{i_p}(p)$, where $1\leqslant i_p\leqslant \dim N_{p}/N_{p-1}$, such that $\{m_{i_k}(k)\colon k<p\} \cup \{m_{i_p}(p)\}$ is a basis of $N_p$. By construction the images of the elements $\{m_{i_p}(p)\}$ form a basis of the quotient $N_p/N_{p-1}$. Then the union $\bigcup\limits_{j=1}^p\{ m_{i_j}(j)\colon 1\leqslant i_j \leqslant \dim N_{j}/N_{j-1}\}$ is a basis of $N_p/N_0$. We use the notation $\widehat{m}_k(p)$ for elements of this union, so that

$$ \begin{equation*} \{\widehat{m}_k(p)\}=\bigcup_{j=1}^p \{ m_{i_j}(j)\colon 1\leqslant i_j \leqslant \dim N_{j}/N_{j-1}\}. \end{equation*} \notag $$

Before considering the case of an arbitrary filtration degree $n$, we consider the cases $n=1,2,3$ separately.

9.2.1. The case $p=1$

By Lemma 4 all elements of $\operatorname{Ext}^{1,*}_{A}(\Bbbk,\Bbbk)$ belong to the filtration $\Phi^1$. By Theorem 2, a non-zero element $a\in \operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)$ for $s>1$ belongs to the image of the differential $d_1\colon N_1/N_0\otimes \operatorname{Ext}^{s-1,*}_{A}(\Bbbk,\Bbbk) \to \operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)$ if and only if it can be represented as a two-fold matric Massey product.

9.2.2. The case $p=2$

Suppose that $a\in \Phi^2\operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)$. Then the image of $a$ in the quotient $\operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)/\operatorname{im}d_1$ is $d_2(y)$ for some $y\in E_2^{2,-s+1,*}$. In the complex $\widetilde{F}(A_*)$, the cochain $y$ is written as

$$ \begin{equation*} \sum_\alpha m_\alpha (2)\otimes [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] + \sum_\beta m_\beta(1) \otimes [\kern-1.5pt[ y_\beta ]\kern-1.5pt] + 1\otimes [\kern-1.5pt[ z ]\kern-1.5pt] , \end{equation*} \notag $$
where $ [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] , [\kern-1.5pt[ y_\beta ]\kern-1.5pt] , [\kern-1.5pt[ z ]\kern-1.5pt] \in F(A_*)$. Therefore, the class $a$ has a representative of the form $dy$ up to addition of an element of the form $d_1\biggl(\,\displaystyle\sum_\beta m_\beta(1) \otimes [\kern-1.5pt[ y'_\beta ]\kern-1.5pt] +1\otimes [\kern-1.5pt[ z' ]\kern-1.5pt] \biggr)$. Moreover, the terms $d(1\otimes [\kern-1.5pt[ z' ]\kern-1.5pt] )=1\otimes d [\kern-1.5pt[ z ]\kern-1.5pt] $ and $d(1\otimes [\kern-1.5pt[ z' ]\kern-1.5pt] )=1\otimes d [\kern-1.5pt[ z ]\kern-1.5pt] $ can be omitted since they change the representative of $a$ by coboundaries. So changing the notation appropriately, we see that the class $a\in \Phi^2\operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)$ has a representative of the form $dy$, where
$$ \begin{equation} y=\sum_\alpha m_\alpha (2)\otimes [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] + \sum_\beta m_\beta(1) \otimes [\kern-1.5pt[ y_\beta ]\kern-1.5pt] . \end{equation} \tag{53} $$
Here $ [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] , [\kern-1.5pt[ y_\beta ]\kern-1.5pt] \in F(A_*)$.

By the definition of the filtration $\{N_p\}$ comultiplication of basis elements $m_\alpha(2)$ is given by

$$ \begin{equation} \Delta m_\alpha(2)=m_\alpha(2)\otimes 1+1\otimes m_\alpha(2)+ \sum_{\beta,\gamma}\theta_\alpha^{\beta\gamma}m_\beta(1)\otimes m_\gamma(1), \end{equation} \tag{54} $$
where $\theta^{\beta\gamma}_\alpha\in\Bbbk$.

Now we write out $dy$ explicitly:

$$ \begin{equation*} dy=U+V, \end{equation*} \notag $$
where
$$ \begin{equation*} U=\sum_\alpha [\kern-1.5pt[ m_\alpha(2)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta [\kern-1.5pt[ m_\beta(1)\otimes y_\beta ]\kern-1.5pt] \end{equation*} \notag $$
and
$$ \begin{equation*} V=\sum_{\alpha,\beta,\gamma}(-1)^{|m_\beta(1)|}\theta_\alpha^{\beta\gamma} m_\beta(1)\otimes [\kern-1.5pt[ m_\gamma(1)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta (-1)^{|m_\beta(1)|}m_\beta(1)\otimes d [\kern-1.5pt[ y_\beta ]\kern-1.5pt] . \end{equation*} \notag $$
We have $V=0$ since $dy$ belongs to the subcomplex $F(A_*)\subset \widetilde{F}(A_*)$. As the elements $m_\beta(1)$ are linearly independent, for each $\beta$ the following equality holds:
$$ \begin{equation} \sum_{\alpha,\gamma}\theta_\alpha^{\beta\gamma} [\kern-1.5pt[ m_\gamma(1)\otimes x_\alpha ]\kern-1.5pt] + d [\kern-1.5pt[ y_\beta ]\kern-1.5pt] =0. \end{equation} \tag{55} $$
Hence the class $a\in \Phi^2\operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)$ is represented by the cochain
$$ \begin{equation*} U=\sum_\alpha [\kern-1.5pt[ m_2(\alpha)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta [\kern-1.5pt[ x_\beta(1)\otimes y_\beta ]\kern-1.5pt] . \end{equation*} \notag $$

Statement 13. (i) For $y\in \widetilde{F}(A_*)$ given by (53) the element $dy$ is a representative of the triple matric Massey product

$$ \begin{equation*} \biggl\langle(\overline{ [\kern-1.5pt[ m_\beta(1) ]\kern-1.5pt] }), \biggl(-\sum_\gamma\theta_\alpha^{\beta\gamma} \overline{ [\kern-1.5pt[ m_\gamma(1) ]\kern-1.5pt] }\biggr), ( [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] )\biggr\rangle \end{equation*} \notag $$
in the cohomology of the complex $F(A_*)$, that is, in $\operatorname{Ext}_A^{*,*}(\Bbbk,\Bbbk)$.

(ii) The Massey product above is non-trivial.

Remark 10. Here $( \overline{ [\kern-1.5pt[ m_\beta(1) ]\kern-1.5pt] } )$ is a row vector, $( [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] ) $ is a column vector, and

$$ \begin{equation*} \biggl(-\displaystyle\sum_\gamma\theta_\alpha^{\beta\gamma} \overline{ [\kern-1.5pt[ m_\gamma(1) ]\kern-1.5pt] }\biggr) \end{equation*} \notag $$
is a matrix.

Proof. Statement (i) is proved by presenting an explicit matric defining system of cochains in $F(A_*)$ for the given Massey product:
$$ \begin{equation*} \begin{alignedat}{3} X(1,1)&=(\overline{ [\kern-1.5pt[ m_\beta(1) ]\kern-1.5pt] })&\quad X(1,2)&=(\overline{ [\kern-1.5pt[ m_\alpha(2) ]\kern-1.5pt] }) && \\ &&\quad X(2,2)&=\biggl(-\sum_\gamma \theta_\alpha^{\beta\gamma} \overline{ [\kern-1.5pt[ m_\gamma(1) ]\kern-1.5pt] }\biggr) &\quad X(2,3)&=( [\kern-1.5pt[ y_\beta ]\kern-1.5pt] ) \\ &&&&\quad X(3,3)&=( [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] ) \end{alignedat} \end{equation*} \notag $$
One of the relations for the defining system, namely
$$ \begin{equation*} d X(1,2)=\overline{X(1,1) }\, X(2,2), \end{equation*} \notag $$
is equivalent to (54). Indeed,
$$ \begin{equation*} \begin{aligned} \, d \overline{ [\kern-1.5pt[ m_\alpha(2) ]\kern-1.5pt] }&=(-1)^{|m_\alpha(2)|+1} d [\kern-1.5pt[ m_\alpha(2) ]\kern-1.5pt] \\ &=(-1)^{|m_\alpha(2)|+1} \sum_{\beta,\gamma} (-1)^{|m_\beta(1)+1|}\theta_\alpha^{\beta\gamma} [\kern-1.5pt[ m_\beta(1) \otimes m_\gamma(1) ]\kern-1.5pt] \\ &=\sum_{\beta,\gamma}(-1)^{|m_\gamma(1)|} \theta_\alpha^{\beta\gamma} [\kern-1.5pt[ m_\beta(1)\otimes m_\gamma(1) ]\kern-1.5pt] \\ &=\sum_\beta\biggl( [\kern-1.5pt[ m_\beta(1) ]\kern-1.5pt] \otimes \biggl(-\sum_\gamma\theta_\alpha^{\beta\gamma} \overline{ [\kern-1.5pt[ m_\gamma(1) ]\kern-1.5pt] }\biggr)\biggr). \end{aligned} \end{equation*} \notag $$

The second relation $d X(2,3)=\overline{X(2,2)}\, X(3,3)$ is obviously equivalent to (55).

The defining system thus constructed defines the cochain

$$ \begin{equation*} \overline{X(1,1)} \, X(2,3) + \overline {X(1,2)}\, X(3,3)= \sum_\alpha [\kern-1.5pt[ m_2(\alpha)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta [\kern-1.5pt[ x_\beta(1)\otimes y_\beta ]\kern-1.5pt] =U, \end{equation*} \notag $$
which represents the class $a$, as shown above.

Now we prove (ii). The same argument applies to the general case. Suppose that there is a defining system with the same $X(1,1)$, $X(2,2)$, $X(3,3)$:

$$ \begin{equation*} \begin{alignedat}{3} &X(1,1) &\quad &Y(1,2) &\quad &Y(1,3) \\ &&\quad &X(2,2) &\quad &Y(2,3) \\ &&&&\quad &X(3,3) \end{alignedat} \end{equation*} \notag $$
where the cochain $Y(1,3)$ satisfies
$$ \begin{equation*} d Y(1,3)=\overline{X(1,1)}Y(2,3)+\overline{Y(1,2)}X(3,3)\quad\text{in}\ \ F(A_*). \end{equation*} \notag $$

The element $Y(1,2)$ is a row of cochains of the form

$$ \begin{equation*} \biggl(\,\displaystyle\sum_{i,p}\lambda_{i,p} [\kern-1.5pt[ m_i(p) ]\kern-1.5pt] \biggr). \end{equation*} \notag $$
The relation $dY(1,2)=\overline{X(1,1)} X(2,2)$ implies that
$$ \begin{equation*} \widetilde{D}\biggl(\,\displaystyle\sum_{i,p}\lambda_{i,p} m_i(p)\biggr)= \displaystyle\sum \theta_{\alpha}^{\beta\gamma}m_\beta(1)\otimes m_\gamma(1). \end{equation*} \notag $$
On the other hand, by the construction of the defining system $X(i,j)$ we have
$$ \begin{equation*} \widetilde D(m_\alpha(2))= \displaystyle\sum\theta_{\alpha}^{\beta\gamma}m_\beta(1)\otimes m_\gamma(1). \end{equation*} \notag $$
By Theorem 2 (c) the map $D$ is injective, which implies that $Y(1,2)=X(1,2)$.

Now we can take $U - dY(1,3)$ instead of $U$ as a cochain representing the class $a$. Then we have

$$ \begin{equation*} \begin{aligned} \, U-dY(1,3)&=\overline{X(1,1)}X(2,3)+\overline{X(1,2)}X(3,3)\\ &\quad - (\overline{X(1,1)}Y(2,3) +\overline{Y(1,2)}X(3,3))\\ &=\overline{X(1,1)}(X(2,3)-Y(2,3)). \end{aligned} \end{equation*} \notag $$
Since $X(1,1)$ is a row vector of basis cocycles, the column vector $X(2,3)-Y(2,3)$ consists of cocycles. It follows that the class represented by the cocycle $U-dY(1,3)$ has filtration 1, in contradiction to the assumption that $a\in \Phi^2$ and $a\notin \Phi^1$. $\Box$

9.2.3. The case $p=3$

Suppose that $a\in \Phi^2 \operatorname{Ext}^{s,*}_A (\Bbbk,\Bbbk)$. As in the previous case, it can be shown that $a$ has a representative of the form $dy$ in $F(A_*)$, where $y\in \widetilde{F}(A_*)$ is given by

$$ \begin{equation} y=\sum_\alpha m_\alpha (3)\otimes [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] + \sum_\beta m_\beta(2) \otimes [\kern-1.5pt[ y_\beta ]\kern-1.5pt] + \sum_\gamma m_\gamma(1) \otimes [\kern-1.5pt[ z_\gamma ]\kern-1.5pt] . \end{equation} \tag{56} $$
Here $ [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] , [\kern-1.5pt[ y_\beta ]\kern-1.5pt] , [\kern-1.5pt[ z_\gamma ]\kern-1.5pt] \in F(A_*)$, and all the $x_\alpha$ are cocycles.

By the definition of the filtration $\{N_p\}$ we have

$$ \begin{equation} \begin{aligned} \, \Delta m_\beta(2) &=1\otimes m_\beta(2) + m_\beta(2) \otimes 1+ \sum_{\gamma,\gamma'}\theta^{\gamma\gamma'}_\beta m_\gamma(1) \otimes m_{\gamma'}(1), \\ \Delta m_\alpha(3) &= 1\otimes m_\alpha(3) + m_\alpha(3) \otimes 1 + \sum_{\beta,\beta'} \theta_\alpha^{\beta \beta'} m_\beta(2)\otimes m_{\beta'}(1) \\ &\qquad+\sum_{\gamma,\gamma''} \theta_\alpha^{\gamma,\gamma''} m_\gamma(1)\otimes \widehat{m}_{\gamma''}(2). \end{aligned} \end{equation} \tag{57} $$
Then
$$ \begin{equation*} dy=U+V+W, \end{equation*} \notag $$
where
$$ \begin{equation*} \begin{aligned} \, U&=\sum_\alpha [\kern-1.5pt[ m_\alpha(3)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta [\kern-1.5pt[ m_\beta(2)\otimes y_\beta ]\kern-1.5pt] + \sum_\gamma [\kern-1.5pt[ m_\gamma(1)\otimes z_\gamma ]\kern-1.5pt] , \\ V &=\sum_{\alpha,\beta,\beta'}(-1)^{|m_\beta(2)|}\theta_\alpha^{\beta\beta'} m_\beta(2)\otimes [\kern-1.5pt[ m_{\beta'}(1)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta (-1)^{|m_\beta(2)|} m_\beta(2)\otimes d [\kern-1.5pt[ y_\beta ]\kern-1.5pt] , \end{aligned} \end{equation*} \notag $$
and
$$ \begin{equation*} \begin{aligned} \, W =\sum_{\alpha,\gamma,\gamma''}(-1)^{|m_\gamma(1)|} \theta_\alpha^{\gamma,\gamma''} m_\gamma(1)\otimes [\kern-1.5pt[ \widehat{m}_{\gamma''}(2)\otimes x_\alpha ]\kern-1.5pt] \\ +\sum_{\beta,\gamma, \gamma'}(-1)^{|m_\gamma(1)|} \theta^{\gamma \gamma'}_\beta m_\gamma(1)\otimes [\kern-1.5pt[ m_{\gamma'}(1)\otimes y_\beta ]\kern-1.5pt] \\ +\sum_{\gamma} (-1)^{|m_\gamma(1)|} m_\gamma(1)\otimes d [\kern-1.5pt[ z_\gamma ]\kern-1.5pt] . \end{aligned} \end{equation*} \notag $$

Since $dy$ belongs to the subcomplex $F(A_*)\subset \widetilde{F}(A_*)$, we obtain $V=0$ and $W=0$. Since the elements $m_\beta(2)$ and $m_\gamma(1)$ are linearly independent, the equalities

$$ \begin{equation} \begin{aligned} \, \sum_{\alpha, \beta'} \theta_\alpha^{\beta\beta'} [\kern-1.5pt[ m_{\beta'}(1)\otimes x_\alpha ]\kern-1.5pt] + d [\kern-1.5pt[ y_\beta ]\kern-1.5pt] &=0, \\ \sum_{\alpha,\gamma''}\theta_\alpha^{\gamma,\gamma''} [\kern-1.5pt[ \widehat{m}_{\gamma''}(2)\otimes x_\alpha ]\kern-1.5pt] + \sum_{\beta, \gamma'} \theta^{\gamma \gamma'}_\beta [\kern-1.5pt[ m_{\gamma'}(1)\otimes y_\beta ]\kern-1.5pt] + d [\kern-1.5pt[ z_\gamma ]\kern-1.5pt] &=0 \end{aligned} \end{equation} \tag{58} $$
hold for each $\beta$ and $\gamma$. Hence $a\in \Phi^2 \operatorname{Ext}^{s,*}_{A}(\Bbbk,\Bbbk)$ is represented by the cochain
$$ \begin{equation*} U=\sum_\alpha [\kern-1.5pt[ m_\alpha(3)\otimes x_\alpha ]\kern-1.5pt] + \sum_\beta [\kern-1.5pt[ m_\beta(2)\otimes y_\beta ]\kern-1.5pt] + \sum_\gamma [\kern-1.5pt[ m_\gamma(1)\otimes z_\gamma ]\kern-1.5pt] . \end{equation*} \notag $$

Statement 14. (i) For $y\in \widetilde{F}(A_*)$ given by (56) the element $dy$ is a representative of the four-fold matric Massey product

$$ \begin{equation*} \biggl\langle(\overline{ [\kern-1.5pt[ m_\gamma(1) ]\kern-1.5pt] }), \biggl(-\sum_{\gamma'} \theta_\beta^{\gamma\gamma'} \overline{ [\kern-1.5pt[ m_{\gamma'}(1) ]\kern-1.5pt] }\biggr), \biggl(-\sum_{\beta'} \theta_\alpha^{\beta\beta'} \overline{ [\kern-1.5pt[ m_{\beta'}(1) ]\kern-1.5pt] }\biggr), ( [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] )\biggr\rangle. \end{equation*} \notag $$

(ii) The Massey product above is non-trivial.

Proof. Statement (i) is proved by presenting an explicit matric defining system of cochains in $F(A_*)$ for the given Massey product:
$$ \begin{equation*} \begin{gathered} \, X(1,1)=(\overline{ [\kern-1.5pt[ m_\gamma(1) ]\kern-1.5pt] }),\qquad X(1,2)=(\overline{ [\kern-1.5pt[ m_\beta(2) ]\kern-1.5pt] }),\qquad X(1,3)=(\overline{ [\kern-1.5pt[ m_\alpha(3) ]\kern-1.5pt] }); \\ X(2,2)=\biggl(-\sum_\gamma \theta_\beta^{\gamma\gamma'} \overline{ [\kern-1.5pt[ m_{\gamma'}(1) ]\kern-1.5pt] }\biggr),\qquad X(2,3)=\biggl(-\sum_{\gamma''}\theta_\alpha^{\gamma,\gamma''} \overline{ [\kern-1.5pt[ \widehat{m}_{\gamma''}(2) ]\kern-1.5pt] }\biggr),\\ X(2,4)=( [\kern-1.5pt[ z_\gamma ]\kern-1.5pt] ); \\ X(3,3)=\biggl(-\sum_{\beta'}\theta_\alpha^{\beta\beta'} \overline{ [\kern-1.5pt[ m_{\beta'}(1) ]\kern-1.5pt] }\biggr),\qquad X(3,4)=( [\kern-1.5pt[ y_\beta ]\kern-1.5pt] );\qquad \\ X(4,4)=( [\kern-1.5pt[ x_\alpha ]\kern-1.5pt] ). \end{gathered} \end{equation*} \notag $$

Two of the relations for the defining system, namely

$$ \begin{equation*} d X(1,2)=\overline{X(1,1)}\, X(2,2)\quad\text{and}\quad d X(1,3)=\overline{X(1,1)}\, X(2,3)+ \overline{X(1,2)}\, X(3,3), \end{equation*} \notag $$
follow from (57).

The two remaining relations

$$ \begin{equation*} d X(2,4)=\overline{X(2,2) }\, X(3,4) + \overline{X(2,3) }\, X(4,4) \quad\text{and}\quad d X(3,4)=\overline{X(3,3) }\, X(4,4) \end{equation*} \notag $$
are obviously equivalent to equalities (58).

The most difficult relation is $dX(2,3)=\overline{X(2,2) }\, X(3,3)$ (relations of this type did not appear for elements of filtration degrees 1 and 2). We derive it from formula (57) and the coassociativity of $\Delta$. In the expression for

$$ \begin{equation*} d X(2,3)=\biggl(-\sum_{\gamma''} \theta_\alpha^{\gamma,\gamma''} d\overline{ [\kern-1.5pt[ \widehat{m}_{\gamma''}(2) ]\kern-1.5pt] }\biggr), \end{equation*} \notag $$
when calculating $d [\kern-1.5pt[ \widehat{m}_{\gamma''}(2) ]\kern-1.5pt] $, we can drop the terms $ [\kern-1.5pt[ {m}_{\gamma''}(1) ]\kern-1.5pt] $ of filtration degree 1, since they all lie in the kernel of $d$. Therefore, only the elements $ [\kern-1.5pt[ {m}_{\gamma''}(2) ]\kern-1.5pt] $ of filtration degree 2 remain. Hence
$$ \begin{equation*} \begin{aligned} \, dX(2,3)&=\biggl(-\sum_{\gamma''}\theta_\alpha^{\gamma,\gamma''} d\overline{ [\kern-1.5pt[ {m}_{\gamma''}(2) ]\kern-1.5pt] }\biggr) \\ &=\biggl(\,\sum_{\gamma'',\gamma',\beta'}(-1)^{|m_{\beta'(1)}|+1} \theta^{\gamma\gamma''}_\alpha\theta_{\gamma''}^{\gamma'\beta'} [\kern-1.5pt[ m_{\gamma'}(1)\otimes m_{\beta'}(1) ]\kern-1.5pt] \biggr). \end{aligned} \end{equation*} \notag $$
On the other hand
$$ \begin{equation*} \overline{X(2,2)}\, X(3,3)=\biggl(\,\sum_{\gamma',\beta, \beta'} \theta_\beta^{\gamma\gamma'}\theta_\alpha^{\beta\beta'} (-1)^{|m_{\beta'}(1)|+1} [\kern-1.5pt[ m_{\gamma'}(1) \otimes m_{\beta'}(1) ]\kern-1.5pt] \biggr). \end{equation*} \notag $$
It remains to check that
$$ \begin{equation} \sum_{\gamma''} \theta^{\gamma\gamma''}_\alpha \theta_{\gamma''}^{\gamma'\beta'}=\sum_\beta\theta_\beta^{\gamma\gamma'} \theta_\alpha^{\beta\beta'}. \end{equation} \tag{59} $$
We use the coassociativity of $\Delta$ and calculate the coefficient of $m_\gamma(1)\otimes m_{\gamma'}(1)\otimes m_{\beta'(1)}$ in
$$ \begin{equation} (\Delta\otimes 1)\circ \Delta m_\alpha(3)=(1\otimes \Delta)\circ \Delta m_\alpha(3). \end{equation} \tag{60} $$
Recall that
$$ \begin{equation} \Delta m_\alpha(3)=1\otimes m_\alpha(3) + m_\alpha(3) \otimes 1 +\!\sum_{\beta,\beta'} \theta_\alpha^{\beta \beta'} m_\beta(2)\otimes m_{\beta'}(1) +\!\sum_{\gamma,\gamma''} \theta_\alpha^{\gamma,\gamma''} m_\gamma(1)\otimes \widehat{m}_{\gamma''}(2). \end{equation} \tag{61} $$
It is easy to see that only the second to the last term on the right-hand side above contributes to the required coefficient in the expression for $(\Delta\otimes 1)\circ \Delta m_\alpha(3)$:
$$ \begin{equation} \begin{aligned} \, \notag (\Delta\otimes 1)\circ \Delta m_\alpha(3)&=\cdots+ \sum_{\beta,\beta'} \theta_\alpha^{\beta \beta'} \Delta(m_\beta(2)) \otimes m_{\beta'}(1) \\ &=\cdots + \sum_{\beta,\beta',\gamma,\gamma'} \theta_\alpha^{\beta \beta'}\theta_\beta^{\gamma\gamma'} m_\gamma(1) \otimes m_{\gamma'}(1)\otimes m_{\beta'}(1). \end{aligned} \end{equation} \tag{62} $$
Here the dots denote elements that are not essential for us. On the other hand only the last term in (61) contributes to the required coefficient in the expression for $(1\otimes \Delta)\circ \Delta m_\alpha(3)$:
$$ \begin{equation} \begin{aligned} \, \notag (1\otimes \Delta)\circ \Delta m_\alpha(3)&=\cdots + \sum_{\gamma,\gamma''} \theta_\alpha^{\gamma,\gamma''} m_\gamma(1) \otimes \Delta\widehat{m}_{\gamma''}(2) \\ \notag &=\cdots+\sum_{\gamma,\gamma''} \theta_\alpha^{\gamma,\gamma''} m_\gamma(1) \otimes \Delta m_{\gamma''}(2) \\ &=\cdots+\sum_{\beta',\gamma,\gamma',\gamma''}\theta_\alpha^{\gamma,\gamma''} \theta_{\gamma''}^{\gamma'\beta'} m_\gamma(1)\otimes m_{\gamma'}(1)\otimes m_{\beta'}(1). \end{aligned} \end{equation} \tag{63} $$
Equating the coefficients of $m_\gamma(1)\otimes m_{\gamma'}(1)\otimes m_{\beta'}(1)$ in (62) and (63), we obtain the required equality (59).

Part (ii) is proved in the same way as for elements of the filtration $\Phi^2$, and we omit this proof. $\Box$

9.2.4. Case $p>3$

The proof in the case of general $k$ is similar to $k=2$ and $k=3$. All the ideas and subtleties of the general case were shown for $k=2,3$.

Let $a\in \Phi^k\operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$. Then there is an element

$$ \begin{equation} y=\sum_{j=1}^k\,\sum_{\alpha_j} m_{\alpha_j}(j)\otimes [\kern-1.5pt[ x_{\alpha_j}(j) ]\kern-1.5pt] \end{equation} \tag{64} $$
in the complex $\widetilde F(A_*)$ such that $dy$ is a representative of $a$ in the complex $F(A_*)$. Here the $m_{\alpha_j}(j)$ are elements of the basis $A_*$ constructed above, the index $\alpha_j$ ranges over some subset of indices from $\{1,\dots,\dim N_j/N_{j-1}\}$, and the $x_{\alpha_j}(j)$ are some cochains from $F(A_*)$. Since $dy$ belongs to the subcomplex $F(A_*)\subset \widetilde F(A_*)$, all elements $x_{\alpha_k}(k)$ are cocycles in $F(A_*)$.

The expansion of $\Delta m_{\alpha_j}(j)$ in the basis selected has the form

$$ \begin{equation} \Delta m_{\alpha_j}(j)=1\otimes m_{\alpha_j}(j)+m_{\alpha_j}(j)\otimes 1+ \sum_{t=1}^{j-1}\,\sum_{\alpha_t ,\alpha_{j,t}} \theta^{\alpha_t \alpha_{j,t}}_{\alpha_j} m_{\alpha_t}(t)\otimes \widehat{m}_{\alpha_{j,t}}(j-t). \end{equation} \tag{65} $$
Note that the summand corresponding to $t=j-1$ has the form
$$ \begin{equation*} \sum_{\alpha_{j-1},\alpha_{j,j-1}} \theta^{\alpha_{j-1},\alpha_{j,j-1}}_{\alpha_j} m_{\alpha_t}(j-1)\otimes m_{\alpha_{j,t}}(1). \end{equation*} \notag $$
Here the second factor of the tensor product has filtration degree exactly $1$, so we can write $m_{\alpha_{j,t}}(1)$ instead of $\widehat m_{\alpha_{j,t}}(1)$.

Now we define matrices $X(i,j)$ consisting of cochains of $F(A_*)$, where $1\leqslant i\leqslant j \leqslant k+1$, $(i,j)\ne (1,k+1)$.

We put

$$ \begin{equation} X(1,j) =(\overline{ [\kern-1.5pt[ m_{\alpha_j}(j) ]\kern-1.5pt] }) \quad\text{(a row vector)}, \end{equation} \tag{66} $$
$$ \begin{equation} X(i+1,k+1) =( [\kern-1.5pt[ x_{\alpha_i}(i) ]\kern-1.5pt] )\quad \text{(a column vector)}, \end{equation} \tag{67} $$
$$ \begin{equation} \nonumber X(t+1,j) =\biggl(-\sum_{\alpha_{j,t}}\theta^{\alpha_t\alpha_{j,t}}_{\alpha_j} \overline{ [\kern-1.5pt[ \widehat m_{\alpha_{j,t}}(j-t) ]\kern-1.5pt] }\biggr) \end{equation} \notag $$
$$ \begin{equation} \qquad\text{for}\ \ 1< t+1 \leqslant j<k+1\quad\text{(a matrix)}. \end{equation} \tag{68} $$
It is easy to see that the diagonal elements have the form
$$ \begin{equation*} X(j,j)=\biggl(\,\sum_{\alpha_{j,j-1}}-\theta^{\alpha_{j-1}, \alpha_{j,j-1}}_{\alpha_j} \overline{ [\kern-1.5pt[ m_{\alpha_{j,t}}(1) ]\kern-1.5pt] }\biggr) \end{equation*} \notag $$
and consist of cocycles, since $d [\kern-1.5pt[ m_{\alpha}(1) ]\kern-1.5pt] = 0$ for any $\alpha$.

Statement 15. (i) The matrices $X(i,j)$, $1\leqslant i\leqslant j\leqslant k+1$, $(i,j)\ne (1,k+1)$, constitute a matric defining system.

(ii) The matric Massey product $\langle X(1,1),X(2,2),\dots,X(k+1,k+1)\rangle$ contains $a\in \operatorname{Ext}^{*,*}_A(\Bbbk,\Bbbk)$.

(iii) The matric Massey product $\langle X(1,1),X(2,2),\dots,X(k+1,k+1)\rangle$ is non-trivial, that is, it does not contain zero.

Proof. To prove (i) consider three cases.

Case 1. The relation $d X(1,j)=\displaystyle\sum_{t=1}^{j-1}\overline{X(1,t)} \, X(t+1,j)$ follows from equality (65) by taking formulae (66) and (68), which define $X(1,t)$ and $X(t+1,j)$ for $1<t+1\leqslant j\leqslant k$, into account.

Case 2. The relation $d X(i,k+1)=\displaystyle\sum_{t=i}^{k} \overline {X(i,t)}\, X(t+1,k+1)$ holds for other reasons. Consider the element $y$ given by (64). Then $dy$ can be written as the sum $U+U_1+U_2+\cdots+ U_{k-1}$, where

$$ \begin{equation} U=\sum_{j,\alpha_j} [\kern-1.5pt[ m_{\alpha_j}(j)\otimes x_{\alpha_j}(j) ]\kern-1.5pt] \end{equation} \tag{69} $$
and
$$ \begin{equation} \nonumber U_{k-1} =\sum_{\alpha_{k-1}}(-1)^{|m_{\alpha_{k-1}}(k-1)|} m_{\alpha_{k-1}}(k-1)\otimes \biggl(d [\kern-1.5pt[ x_{\alpha_{k-1}}(k-1) ]\kern-1.5pt] \end{equation} \notag $$
$$ \begin{equation} \qquad+\sum_{\alpha_k, \alpha_{k,k-1}} \theta_{\alpha_k}^{\alpha_{k-1}, \alpha_{k,k-1}} [\kern-1.5pt[ m_{\alpha_{k,k-1}}(1)\otimes x_{\alpha_k}(k) ]\kern-1.5pt] \biggr), \end{equation} \tag{70} $$
$$ \begin{equation} \nonumber U_{k-2} =\sum_{\alpha_{k-2}}(-1)^{|m_{\alpha_{k-2}}(k-2)|} m_{\alpha_{k-2}}(k-2)\otimes \biggl(d [\kern-1.5pt[ x_{\alpha_{k-2}}(k-2) ]\kern-1.5pt] \end{equation} \notag $$
$$ \begin{equation} \nonumber \qquad + \sum_{\alpha_{k-1}, \alpha_{k-1,k-2}} \theta_{\alpha_{k-1}}^{\alpha_{k-2},\alpha_{k-1,k-2}} [\kern-1.5pt[ m_{\alpha_{k-1,k-2}}(1)\otimes x_{\alpha_{k-1}}(k-1) ]\kern-1.5pt] \end{equation} \notag $$
$$ \begin{equation} \qquad +\sum_{\alpha_{k}, \alpha_{k,k-2}} \theta_{\alpha_{k}}^{\alpha_{k-2},\alpha_{k,k-2}} [\kern-1.5pt[ \widehat{m}_{\alpha_{k,k-2}}(2)\otimes x_{\alpha_{k}}(k) ]\kern-1.5pt] \biggr), \end{equation} \tag{71} $$
$$ \begin{equation} \nonumber \qquad\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \ldots\ldots\ldots\ldots\ldots \end{equation} \notag $$
$$ \begin{equation} \nonumber U_1= \sum_{\alpha_{1}}(-1)^{|m_{\alpha_{1}}(1)|} m_{\alpha_{1}}(1) \end{equation} \notag $$
$$ \begin{equation} \qquad\otimes\biggl( d [\kern-1.5pt[ x_{\alpha_{1}}(1) ]\kern-1.5pt] +\sum_{t=2}^k\, \sum_{\alpha_t,\alpha_{t,1}} \theta_{\alpha_t}^{\alpha_1,\alpha_{t,1}} [\kern-1.5pt[ \widehat m_{\alpha_{t,1}}(t-1)\otimes x_{\alpha_t}(1) ]\kern-1.5pt] \biggr) \end{equation} \tag{72} $$
or, more generally,
$$ \begin{equation} \begin{aligned} \, \notag U_s&=\sum_{\alpha_{s}}(-1)^{|m_{\alpha_{s}}(s)|}m_{\alpha_{s}}(s)\otimes \biggl( d [\kern-1.5pt[ x_{\alpha_{s}}(s) ]\kern-1.5pt] \\ &\qquad+\sum_{t=s+1}^k\, \sum_{\alpha_t,\alpha_{t,s}}\theta_{\alpha_t}^{\alpha_s,\alpha_{t,s}} [\kern-1.5pt[ \widehat m_{\alpha_{t,s}}(t-s)\otimes x_{\alpha_t}(s) ]\kern-1.5pt] \biggr). \end{aligned} \end{equation} \tag{73} $$

Since $dy$ belongs to the subcomplex $F(A_*)\subset\widetilde F(A_*)$, all elements $U_1,\dots,U_{k-1}$ are zero. The linear independence of the elements $m_{\alpha_{s}}(s)$ implies the relations

$$ \begin{equation*} d [\kern-1.5pt[ x_{\alpha_{s}}(s) ]\kern-1.5pt] +\sum_{t=s+1}^k\, \sum_{\alpha_t,\alpha_{t,s}}\theta_{\alpha_t}^{\alpha_s,\alpha_{t,s}} [\kern-1.5pt[ \widehat m_{\alpha_{t,s}}(t-s)\otimes x_{\alpha_t}(s) ]\kern-1.5pt] =0 \end{equation*} \notag $$
for all $s$ and $\alpha_s$. It follows that
$$ \begin{equation*} dX(s+1,k+1)=\sum_{t=s+1}^{k}\overline{X(s+1,t)} X(t+1,k+1). \end{equation*} \notag $$

Case 3. The relations for $dX(i,j)$ for $i>1$ and $j<k+1$ are derived from (68) and the associativity of comultiplication. In this case the argument is the same as in the proof of the relation for $dX(2,3)$ in Statement 14.

To prove (ii) recall that we proved before the equality $dy= U$, where the cochain $U\in F(A_*)$ is given by (69). On the other hand it is easy to see that the cocycle $\displaystyle\sum_{t=1}^k\overline{X(1,t)}\, X(t+1,k+1)$ representing the Massey product under consideration coincides with $U$.

Finally, we prove (iii). We argue by contradiction: suppose that there exists a defining system $Y(i,j)$ such that $Y(i,i)=X(i,i)$ for all $i=1,\dots,k+1$ and such that the cocycle $\displaystyle\sum_{t=1}^k \overline{Y(1,t)}\, Y(t+1,k+1)$ is the coboundary of some element, which we denote by $Y(1,k+1)$.

We prove by induction that for all $t$ the element $Y(1,t)$ differs from $X(1,t)$ by summands of the form $\delta(1,t)=\biggl(\,\displaystyle\sum_{i;\,p<t}c_{\mu,i,p}m_i(p)\biggr)$, that is, by elements of filtration degree strictly less than $t$. For $Y(1,1)= X(1,1)$ this is obviously true.

By induction consider the row vector $Y(1,t+1)$. It consists of elements of the form $\displaystyle\sum_{i;\,p<t}c_{\mu,i,p}m_i(p)$, where $\mu$ is the index of the element in the row and there are no a priori restrictions on $p$. From the defining relation and the induction hypothesis we have

$$ \begin{equation*} d(Y(1,t+1))=\sum_{s=1}^{t}\overline{Y(1,s)}\,Y(s+1,t+1)= \sum_{s=1}^{t}\overline{(X(1,s)+\delta(1,s))}\,Y(s+1,t+1). \end{equation*} \notag $$
We expand the right-hand side with respect to the natural basis of $F(A_*)$, which consists of tensor products of elements $m_i(j)$. In this expansion there are no terms beginning with $m_{\alpha_p}(p)$ for $p>t$, and the terms beginning with $m_{\alpha_t}(t)$ come from $X(1,t)X(t+1,t+1)$ and therefore have the form
$$ \begin{equation*} \sum_{\alpha_t,\alpha_{t+1,t}}\theta_{\alpha_{t+1}}^{\alpha_t,\alpha_{t+1,t}} m_{\alpha_t}(t)\otimes m_{\alpha_{t+1,t}}=\widetilde D(m_{\alpha_{t+1}}(t+1)). \end{equation*} \notag $$
Since $\widetilde D$ is injective, there are no elements of filtration degree greater than $t+1$ in $Y(1,t+1)$, and the elements of filtration degree $t$ appear with the same coefficients as in $X(1,t+1)$.

Thus, $d Y(1,k+1)=\overline{X(1,k)}\,X(k+1,k+1)+T$, where $T$ denotes elements of filtration less than $k$. Hence the cohomology class $a$ has a representative $U- d Y(1,k+ 1)$ of filtration degree less than $k$, which contradicts the assumption that the filtration degree of $a$ is precisely $k$. $\Box$

The authors are grateful to Professor I. K. Babenko for his suggestions and comments, which led to improvements in the presentation of our results.


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Citation: V. M. Buchstaber, F. Yu. Popelenskii, “Cohomology of Hopf algebras and Massey products”, Russian Math. Surveys, 79:4 (2024), 567–648
Citation in format AMSBIB
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\paper Cohomology of Hopf algebras and Massey products
\jour Russian Math. Surveys
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\vol 79
\issue 4
\pages 567--648
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