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 Uspekhi Mat. Nauk, 2017, Volume 72, Issue 6(438), Pages 139–190 (Mi umn9802)

On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

L. O. Chekhova, M. Mazzoccob

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Loughborough University, Loughborough, UK

Abstract: Let $\mathscr A$ be the space of bilinear forms on $\mathbb C^N$ with defining matrices $\mathbb A$ endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action $\mathbb A\mapsto B\mathbb AB^{\mathrm{T}}$ of the $\mathrm{GL}_N$ Poisson–Lie group on $\mathscr A$. A classification is given of all possible quadratic brackets on $(B,\mathbb A)\in \mathrm{GL}_N\times \mathscr A$ preserving the Poisson property of the action, thus endowing $\mathscr A$ with the structure of a Poisson homogeneous space. Besides the product Poisson structure on $\mathrm{GL}_N\times \mathscr A$, there are two other (mutually dual) structures, which (unlike the product Poisson structure) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples $(B,C,\mathbb A)\in \mathrm{GL}_N\times \mathrm{GL}_N\times \mathscr A$ with the Poisson action $\mathbb A\mapsto B\mathbb AC^{\mathrm{T}}$, and it is shown that $\mathscr A$ then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid.
Bibliography: 30 titles.

Keywords: of bilinear forms, Poisson–Lie action, block upper triangular matrices, quantum algebras, central elements, Dirac reduction, groupoid.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 Engineering and Physical Sciences Research Council EP/J007234/1 Sections 8 and 9 of this paper were written by L. O. Chekhov, and §§ 4–7 and 10–12 were written by M. Mazzzocco. Chekhov's research was supported by the Russian Science Foundation under grant no. 14-50-00005 at the Steklov Mathematical Institute of Russian Academy of Sciences. Mazzocco's research was supported by the Engineering and Physical Sciences Research Council of Great Britain (grant no. EP/J007234/1).

DOI: https://doi.org/10.4213/rm9802

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English version:
Russian Mathematical Surveys, 2017, 72:6, 1109–1156

Bibliographic databases:

ArXiv: 1404.0988
Document Type: Article
UDC: 514.7+512.548
MSC: Primary 53D17; Secondary 16T25, 20L05

Citation: L. O. Chekhov, M. Mazzocco, “On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action”, Uspekhi Mat. Nauk, 72:6(438) (2017), 139–190; Russian Math. Surveys, 72:6 (2017), 1109–1156

Citation in format AMSBIB
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