Lifting automorphisms of quotients of adjoint representations.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1337.20051
author: Gerald W. Schwarz
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.6300
full work available at URL: jlt24026.htm
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 6365974
zbMATH Keywords: differential operators
zbMATH Keywords: automorphisms
zbMATH Keywords: quotients
zbMATH Keywords: adjoint representations
zbMATH Keywords: simple complex Lie algebras
zbMATH Keywords: actions of compact Lie groups
MaRDI profile type: Publication
arXiv ID: 1301.6300
arXiv classification: math.RT
arXiv classification: math.GR
title: Lifting automorphisms of quotients of adjoint representations.
published in: Journal of Lie Theory
review text: Let \(\mathfrak g_i\) be simple complex Lie algebras, \(1\leq i\leq d\), and let \(G=G_1\times\cdots\times G_d\) be the corresponding adjoint group. Consider the \(G\)-module \(V=\bigoplus r_ i\mathfrak g_ i\) where \(r_i\in\mathbb N\) for all \(i\). The author says that \(V\) is \(large\) if \(r_i\geq 2\) for all \(i\) and \(r_i\geq 3\) when \(G_ i\) has rank 1. Earlier [J. Lond. Math. Soc., II. Ser. 89, No. 1, 169-193 (2014; Zbl 1339.20038)], the author showed that when \(V\) is large any algebraic automorphism \(\psi\) of the quotient \(Z:=V//G\) lifts to an algebraic mapping \(\Psi\colon V\to V\) which sends the fiber over \(z\) to the fiber over \(\psi(z)\) when \(z\in Z\). Now he obtains a similar result for holomorphic \(\psi\). He determines which \(\psi\) lift if \(V\) is not large. A similar problem is considered for actions of compact Lie groups.
reviewed by: L. N. Vaserstein
description: scientific article; zbMATH DE number 6365974