Poisson structures transverse to coadjoint orbits.
type: Item
instance of: scholarly article
title: Poisson structures transverse to coadjoint orbits.
zbMATH Open document ID: 1074.53070
DOI: 10.1016/S0007-4497(02)01118-1
author: Richard Cushman
author: R. M. Roberts
published in: Bulletin des Sciences Mathématiques
publication date: 2003-03-23T00:00:00Z
review text: The authors prove a conjecture by \textit{P. Damianou} [ibid. 120, No. 2, 195--214 (1996; Zbl 0853.58048)], that the transverse Poisson structure to a coadjoint orbit of a semisimple, real, Lie algebra is polynomial. The precise statement is as follows. Let \(\mathfrak{g}\) be a semisimple, real, Lie algebra. Let \(\langle\cdot,\cdot\rangle\) denote the Killing form, and fix \(\nu\in\mathfrak{g}^*\). Also, let \(\mathfrak{g}_\nu\) be the isotropy Lie algebra of \(\nu\) and denote by \(\mathfrak{n}\) its orthogonal complement, so that \( \mathfrak{g}=\mathfrak{g}_\nu\oplus \mathfrak{n}.\) This gives rise to a dual decomposition \( \mathfrak{g}^*=\mathfrak{g}_\nu^0\oplus \mathfrak{n}^0.\) The affine space \(N=\nu+\mathfrak{n}^0\) is transverse at \(\nu\) to the coadjoint orbit through \(\nu\), hence, carries a Poisson structure in a neighborhood of \(\nu\). The authors show that this Poisson structure is polynomial.
reviewed by: Rui Loja Fernandes
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 1886864
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: transverse structure
MaRDI profile type: Publication
cites work: Q5628386
cites work: Transverse Poisson structures of coadjoint orbits
cites work: Q5587174
cites work: Q4156946
cites work: Q5690465
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Some remarks on the transverse Poisson structures of coadjoint orbits
cites work: Hamiltonian systems near relative equilibria
cites work: Poisson algebras and transverse structures
cites work: Simple singularities and simple algebraic groups
cites work: Errata and addenda
description: scientific article; zbMATH DE number 1886864