The derived algebra of a stabilizer, families of coadjoint orbits, and sheets
type: Item
instance of: scholarly article
zbMATH Open document ID: 1373.17015
author: Anton Izosimov
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.1135
full work available at URL: jlt24030.htm
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 6365978
zbMATH Keywords: Lie algebra
zbMATH Keywords: coadjoint action
zbMATH Keywords: derived algebra of stabilizer
zbMATH Keywords: tangent space
MaRDI profile type: Publication
arXiv ID: 1202.1135
arXiv classification: math.RT
title: The derived algebra of a stabilizer, families of coadjoint orbits, and sheets
published in: Journal of Lie Theory
review text: Let \(L\) be a finite dimensional real or complex Lie algebra and let \(\mu\) be an element of the dual space, \(L^*\). The adjoint group of \(L\) acts on \(L^*\) by the coadjoint action. Let \(L_k^*\) be the union of all orbits of codimension k under this action. Further, let \(L_{\mu} = \{x\in L \mid \text{ad}_x^* (\mu)=0\}\), \(\mu\in L^*\). Then any \(\alpha \in L^*\) which is tangent to \(L_k^*\) at a point \(\mu \in L^*\) vanishes on the derived algebra of \(L_{\mu}\). It follows that each element of the tangent space vanishes on the derived algebra of \(L_{\mu}\) and the dimension of this derived algebra is less than or equal to the codimension of \(L_k^*\) where \(k\) is the dimension of \(L_{\mu}\). Conditions equivalent to equality in the last statement are shown.
reviewed by: Ernest L. Stitzinger
description: scientific article; zbMATH DE number 6365978