Nondegenerate Lie algebras
type: Item
instance of: scholarly article
title: Nondegenerate Lie algebras
zbMATH Open document ID: 0886.58041
DOI: 10.1007/BF02362634
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-01-06T00:00:00Z
review text: For a Poisson structure \(\{.,.\}\) on a manifold \(M\) we choose the local coordinates \((x_1,\ldots,x_n)\) vanishing at fixed \(x_0\in M\) and consider the Taylor expansion \(\{x_i,x_j\}=\sum_kC^k_{ij}x_k+\ldots\). Since \(\{.,.\}\) satisfies the Jacobi identity, the \(C_{ij}^k\) are structure constants of a well-defined Lie algebra \(g\) which is called the linear part of \(\{.,.\}\). A finite dimensional Lie algebra \(g\) is called \textit{nondegenerate} if for every Poisson structure which vanishes at the point \(x_0\in M\) and which has \(g\) as linear part, there exist new coordinates \((x_1',\ldots,x'_n)\) with \(\{x'_i,x'_j\}=\sum_kC^k_{ij}x'_k\). In addition to the ``nondegeneracy'' defined above, we can also consider formal nondegeneracy (where Poisson structures and changes of coordinates are formal) and analytic nondegeneracy (where Poisson structures and changes of coordinates are analytical). \textit{A. Weinstein} [J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011)] proved that every semisimple Lie algebra is formally nondegenerate. \textit{J. F. Conn} proved in Ann. Math. 119, 577-601 (1984; Zbl 0553.58004) that every semisimple Lie algebra is analytically nondegenerate and in Ann. Math. 121, 565-593 (1985; Zbl 0592.58025) that every compact semisimple Lie algebra is \(C^\infty\)-nondegenerate. Let \(A_2\) be a nontrivial 2-dimensional Lie algebra. The main results of the paper are Proposition 1. If \(g\) is a semisimple Lie algebra, then the semidirect product \(g\propto\mathbb{R}\) is formally and analytically nondegenerate. If \(g\) is semisimple compact, then \(g\propto\mathbb{R}\) is \(C^{\infty}\)-nondegenerate. Proposition 2. The product \(A_2\propto\ldots\propto A_2\) of \(n\) copies of \(A_2\) is formally and analytically nondegenerate. \(A_2\propto A_2\) is \(C^\infty\)-nondegenerate.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 1017600
zbMATH Keywords: Poisson structure
zbMATH Keywords: \(C^ \infty\)-nondegenerate Lie algebra
zbMATH Keywords: formally nondegenerate Lie algebra
zbMATH Keywords: analytically nondegenerate Lie algebra
author: Jean-Paul Dufour
reviewed by: Mircea Crasmareanu
MaRDI profile type: Publication
cites work: On the solutions of analytic equations
cites work: Normal forms for analytic Poisson structures
cites work: Normal forms for smooth Poisson structures
cites work: Linéarisation de certaines structures de Poisson. (Linearization of certain Poisson structures)
cites work: The local structure of Poisson manifolds
cites work: Poisson geometry of the principal series and nonlinearizable structures
full work available at URL: https://doi.org/10.1007/bf02362634
OpenAlex ID: W2093230768
Recommended article: Q4526543
Recommended article: Q4011291
Recommended article: A family of degenerate Lie algebras
Recommended article: Connections with prescribed curvature and Yang-Mills currents : the semi-simple case
Recommended article: A new family of nondegenerate Lie algebras
Recommended article: A manifold structure for analytic isotropy Lie pseudogroups of infinite type
Recommended article: Q3764914
Recommended article: Normalisation formelle de structures de Poisson
Recommended article: Nondegeneracy of the Lie algebra \(\text{aff}(n)\).
Recommended article: Levi decomposition of analytic Poisson structures and Lie algebroids.
description: scientific article; zbMATH DE number 1017600