Constants of Lie algebra actions
type: Item
instance of: scholarly article
title: Constants of Lie algebra actions
zbMATH Open document ID: 0653.16025
DOI: 10.1016/0021-8693(88)90303-1
author: Jeffrey Bergen
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let R be an associative algebra (not necessarily with 1) over a field k. Section 1 of the present article considers actions of finite-dimensional nilpotent Lie algebras L by algebraic derivations on R. The main result states that if R is non-nilpotent then \(R^ L=\{r\in R|\) \(\ell \circ r=0\) for all \(\ell \in L\}\), the ring of constants of L on R, is non- zero. Section 2 deals, more generally, with actions of Hopf algebras H on R. This covers the case of Lie algebra actions by taking H to be the enveloping algebra of L (restricted when L is restricted) as well as the case of group actions, with H being the group algebra. Under a suitable technical assumption on H, the author constructs a non-nilpotent algebra R on which H acts with zero constants, \(R^ H=\{0\}\). This justifies in particular the nilpotency assumption on L in Section 1. The article concludes with a number of open questions.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 4065213
zbMATH Keywords: actions of finite-dimensional nilpotent Lie algebras
zbMATH Keywords: algebraic derivations
zbMATH Keywords: ring of constants
zbMATH Keywords: actions of Hopf algebras
zbMATH Keywords: Lie algebra actions
zbMATH Keywords: enveloping algebra
zbMATH Keywords: group actions
zbMATH Keywords: group algebra
reviewed by: Martin Lorenz
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90303-1
OpenAlex ID: W1973746044
Wikidata QID: Q115366303
cites work: Smash products and outer derivations
cites work: Actions of Commutative Hopf Algebras
cites work: Rings with Fixed-Point-Free Group Actions
cites work: Hopf algebra actions
cites work: Group-Graded Rings, Smash Products, and Group Actions
cites work: Group graded rings
cites work: CONSTANTS OF DERIVATIONS OF PRIME RINGS
cites work: Q5639810
cites work: Centralizers in rings
cites work: Representations of Restricted Lie Algebras of Characteristic p
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Observations on group rings
cites work: Rings with an integral element whose centralizer satisfies a polynomial identity
cites work: Q5590089
description: scientific article; zbMATH DE number 4065213