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Mat. Sb., 2019, Volume 210, Number 6, Pages 111–160 (Mi msb9055)  

Naturally graded Lie algebras of slow growth

D. V. Millionshchikovab

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: A pro-nilpotent Lie algebra $\mathfrak g$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra $\operatorname{gr}\mathfrak g$ with respect to the filtration by the ideals in the lower central series. Finite-dimensional naturally graded Lie algebras are known in sub-Riemannian geometry and geometric control theory, where they are called Carnot algebras.
We classify the finite-dimensional and infinite-dimensional naturally graded Lie algebras $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ with the property
$$ \dim\mathfrak g_i+\dim\mathfrak g_{i+1}\le3,\qquad i\ge1. $$
An arbitrary Lie algebra $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ of this class is generated by the two-dimensional subspace $\mathfrak g_1$, and the corresponding growth function $F_\mathfrak g^\mathrm{gr}(n)$ satisfies the bound $F_\mathfrak g^\mathrm{gr}(n)\le3n/2+1$.
Bibliography: 32 titles.

Keywords: graded Lie algebra, Carnot algebra, Kac-Moody algebras, central extension, automorphism.

Funding Agency Grant Number
Russian Science Foundation 14-11-00414
This work was supported by the Russian Science Foundation under grant 14-11-00414.


DOI: https://doi.org/10.4213/sm9055

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English version:
Sbornik: Mathematics, 2019, 210:6, 862–909

UDC: 512.812.4
MSC: 17B30
Received: 27.12.2017 and 31.05.2018

Citation: D. V. Millionshchikov, “Naturally graded Lie algebras of slow growth”, Mat. Sb., 210:6 (2019), 111–160; Sb. Math., 210:6 (2019), 862–909

Citation in format AMSBIB
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\by D.~V.~Millionshchikov
\paper Naturally graded Lie algebras of slow growth
\jour Mat. Sb.
\yr 2019
\vol 210
\issue 6
\pages 111--160
\mathnet{http://mi.mathnet.ru/msb9055}
\crossref{https://doi.org/10.4213/sm9055}
\elib{http://elibrary.ru/item.asp?id=37652220}
\transl
\jour Sb. Math.
\yr 2019
\vol 210
\issue 6
\pages 862--909
\crossref{https://doi.org/10.1070/SM9055}


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