Left invariant Poisson structures on classical non-compact simple Lie groups
type: Item
instance of: scholarly article
title: Left invariant Poisson structures on classical non-compact simple Lie groups
zbMATH Open document ID: 0951.22010
DOI: 10.1007/BF02773218
author: Yoshio Agaoka
published in: Israel Journal of Mathematics
publication date: 2000-09-19T00:00:00Z
review text: Let \(\{\;,\;\}\) be a Poisson structure on a differentiable manifold \(M\) and let \(\pi \in{\Gamma}({\wedge}^2TM)\) be the contravariant 2-tensor field on \(M\) defined by \(\langle \pi,df{\wedge}dg\rangle=\{f,g\}\), \(f,g{\in}C^{\infty}(M)\), where \(\langle\;,\;\rangle\) is the natural pairing [see \textit{I. Vaisman}, Lectures on the geometry of Poisson manifolds, Prog. Math. 118 ( Basel, 1994; Zbl 0810.53019)]. In the case when \(M\) is a Lie group \(G\), a Poisson structure on \(G\) is called left invariant if \(\pi\) is a left invariant 2-tensor field, i.e., \(\pi\) belongs to the space \({\wedge}^2g\), where \(g\) is the Lie algebra of \(G\), consisting of left invariant vector fields. The main purpose of the present paper is to give new examples of high rank left invariant Poisson structures for all non-compact classical real simple Lie groups. The result is equivalent to giving constant solutions of the classical Yang-Baxter equation \([r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13}, r^{23}]=0\), where \(r\) takes values in the space \({\wedge}^2g\) for these Lie groups.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 1445780
zbMATH Keywords: Poisson structure
zbMATH Keywords: Lie group
zbMATH Keywords: left invariant Poisson structure
zbMATH Keywords: real simple Lie groups
zbMATH Keywords: Yang-Baxter equation
Wikidata QID: Q115391653
reviewed by: Neculai Papaghiuc
MaRDI profile type: Publication
cites work: Uniqueness of left invariant symplectic structures on the affine Lie group
cites work: Solutions of the classical Yang-Baxter equation for simple Lie algebras
cites work: Q3346994
cites work: Q4341429
cites work: The affine group as symplectic manifold
cites work: Parameter-dependent solutions of the classical Yang-Baxter equation on \(sl(n,\mathbb{C})\)
cites work: Q4328349
cites work: Graded Lie Algebras of the Second Kind
cites work: Symplectic Homogeneous Spaces
cites work: Q3040173
cites work: The maximal subgroups of the classical groups
cites work: Q3996563
cites work: Q3897665
cites work: On the subalgebras \({\mathfrak g}_ 0\) and \({\mathfrak g}_{ev}\) of semisimple graded Lie algebras
cites work: Q5517524
cites work: On Lie groups with left-invariant symplectic or Kählerian structures
cites work: On quadratic Poisson structures
cites work: Poisson Lie groups, dressing transformations, and Bruhat decompositions
cites work: Q3973392
cites work: Q4328358
cites work: Godbillon-Vey classes of symplectic foliations
cites work: The maximal solvable subgroups of the SU(p,q) groups and all subgroups of SU(2,1)
cites work: Constant solutions of Yang-Baxter equation for $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$.
cites work: Lectures on the geometry of Poisson manifolds
cites work: Poisson structures on the Lorentz group
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description: scientific article; zbMATH DE number 1445780