Invariants for the action of a finite group on the enveloping algebra of a semisimple Lie algebra
type: Item
instance of: scholarly article
title: Invariants for the action of a finite group on the enveloping algebra of a semisimple Lie algebra
zbMATH Open document ID: 0849.16035
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: Let \(G\) be a simply connected semisimple group and \(U\) the enveloping algebra of its Lie algebra (over \(\mathbb{C}\)). The object of this article is to study, for any finite subgroup \(H\) of \(G\), the algebra \(R=U^H\) of invariants under the natural action of \(H\) on \(U\), in particular the finite-dimensional irreducible \(R\)-modules and their annihilators. By means of the standard Morita context between \(R\) and the skew group ring \(S=U*H\), every (finite-dimensional, irreducible) \(R\)-module \(V\) can be written as the submodule \(W^H\) of \(H\)-invariants of some (finite-dimensional, irreducible) \(S\)-module \(W\). Thus it suffices to describe the finite-dimensional irreducible \(S\)-modules \(W\). These are shown to be of the form \(W=M\otimes_\mathbb{C} X\), where \(M\) is an irreducible \(U\)-module and \(X\) is an irreducible representation of \(H\). Especially complete results are obtained for \(G=\text{SL}_2\), for which the finite subgroups \(H\) are known. In each case, the McKay correspondence is used to explicitly calculate the number of non-isomorphic irreducible \(R\)-modules of a given dimension. It turns out that this number is always independent of the chosen dimension. Much of this article is motivated by and elaborates on general results of \textit{H. Kraft} and \textit{L. W. Small} [Math. Res. Lett. 1, No. 3, 297-307 (1994; see the following review Zbl 0849.16036)].
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 883793
zbMATH Keywords: algebra of invariants
zbMATH Keywords: simply connected semisimple groups
zbMATH Keywords: enveloping algebras
zbMATH Keywords: Lie algebras
zbMATH Keywords: finite subgroups
zbMATH Keywords: finite-dimensional irreducible modules
zbMATH Keywords: annihilators
zbMATH Keywords: skew group rings
zbMATH Keywords: McKay correspondence
Wikidata QID: Q115396097
author: Philippe Caldero
author: Gadi Shimon Perets
reviewed by: Martin Lorenz
MaRDI profile type: Publication
full work available at URL: jabr.1996.0153
OpenAlex ID: W1983735693
DOI: 10.1006/JABR.1996.0153
Recommended article: Isomorphisms of finite invariant for enveloping algebras, semi-simple case
Recommended article: Invariants for automorphisms of certain iterated skew polynomial rings
Recommended article: ON REPRESENTATIONS OF ALGEBRAIC GROUPS IN CHARACTERISTIC TWO
Recommended article: Q3158968
Recommended article: Invariants of centralisers in positive characteristic
Recommended article: Q4878360
Recommended article: Invariant Differential Operators and FCR Factors of Enveloping Algebras
Recommended article: Q3575635
Recommended article: Q3395889
Recommended article: Exterior Powers of the Adjoint Representation
description: scientific article; zbMATH DE number 883793