A rigidity theorem for finite group actions on enveloping algebras of semisimple Lie algebras
type: Item
instance of: scholarly article
title: A rigidity theorem for finite group actions on enveloping algebras of semisimple Lie algebras
zbMATH Open document ID: 0823.17011
author: Jacques Alev
author: P. Polo
published in: Advances in Mathematics
publication date: 1995-05-30T00:00:00Z
review text: Let \(G\) be a finite group of automorphisms acting (not necessarily linearly) on the enveloping algebra \(U ({\mathfrak g})\) of a semisimple Lie algebra \({\mathfrak g}\) over an algebraically closed field \(k\) of characteristic zero. The main result of this paper is that if the fixed subalgebra \(U^ G\) is \(k\)-isomorphic to any enveloping algebra, then \(G\) is trivial; the same result is also shown for the \(n\)th Weyl algebra \(A_ n (k)\). In other words, enveloping algebras of semisimple Lie algebras and Weyl algebras do not admit Galois embeddings into themselves; note that symmetric algebras by contrast do admit such embeddings. The proof of the first result uses certain very precise but quite classical facts about primitive ideals in \(U ({\mathfrak g})\); it actually shows more generally that \(U^ G\) cannot even be a quotient of an enveloping algebra if \(G\) acts trivially on the center of \(U ({\mathfrak g})\) (and \(G \neq 1)\). The second result is proved in two ways, one of which involves passing to prime characteristic and the other higher \(K\)-theory. The second proof is much shorter but of course less elementary and unfortunately applies only to linear actions.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 759446
zbMATH Keywords: rigidity theorem
zbMATH Keywords: automorphisms
zbMATH Keywords: enveloping algebras
zbMATH Keywords: semisimple Lie algebras
zbMATH Keywords: Weyl algebras
zbMATH Keywords: Galois embeddings
Wikidata QID: Q115396411
reviewed by: William M. McGovern
MaRDI profile type: Publication
full work available at URL: aima.1995.1022
OpenAlex ID: W2075599432
DOI: 10.1006/AIMA.1995.1022
description: scientific article; zbMATH DE number 759446