On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups
type: Item
instance of: scholarly article
title: On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups
zbMATH Open document ID: 1327.22008
published in: Journal of Mathematical Sciences (New York)
publication date: 2015-10-29T00:00:00Z
review text: Given a Lie group \(G\), the famous Kirillov-Kostant-Souriau's theorem asserts that each co-adjoint orbit of \(\lambda\in {\mathfrak g}^*\) carries a natural symplectic structure that is invariant under \(G\). The importance of co-adjoint representations was emphasized by \textit{A. A. Kirillov} [Bull. Am. Math. Soc., New Ser. 36, No. 4, 433--488 (1999; Zbl 0940.22013); Lectures on the orbit method. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1229.22003)], Kostant, Auslander, Pukánszky and others. Also see [\textit{D. A. Vogan Jr.}, ``Book review of: A. A. Kirillov, Lectures on the orbit method'', Bull. Am. Math. Soc. 42, No. 4, 535--544 (2005)]. \textit{I. M. Gel'fand} and \textit{M. A. Naĭmark} [Tr. Mat. Inst. Steklova 36, 1--288 (1950; Zbl 0041.36206)] introduced the method of canonical parametrization of co-adjoint orbits of the general linear group. In the paper under review, the method is extended to matrix groups preserving a bilinear quadratic form. The method is insensitive to the Jordan form of the orbit and at each step it turns from the matrix of a transformation \(A\) to the matrix of the transformation that is the projection of \(A\) parallel to an eigenspace of this transformation to a coordinate subspace.
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 17B08
zbMATH DE Number: 6501791
zbMATH Keywords: co-adjoint orbit
zbMATH Keywords: Jordan form
zbMATH Keywords: parametrization
author: Mikhail Vasilievich Babich
reviewed by: Tin-Yau Tam
Wikidata QID: Q115382325
MaRDI profile type: Publication
full work available at URL: s10958-015-2530-2
OpenAlex ID: W2197059584
cites work: Q5802185
cites work: \(\mathcal R\)-matrix and Baxter \(\mathcal Q\)-operators for the noncompact \(\mathrm{SL}(N,\mathbb C)\) invariant spin chain
cites work: On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case
cites work: UNITARY REPRESENTATIONS OF NILPOTENT LIE GROUPS
cites work: Q4167627
cites work: Q5690465
cites work: Q3029731
DOI: 10.1007/S10958-015-2530-2
description: scientific article; zbMATH DE number 6501791