Invariants of Lie superalgebras acting on associative algebras
type: Item
instance of: scholarly article
title: Invariants of Lie superalgebras acting on associative algebras
zbMATH Open document ID: 0863.16028
DOI: 10.1007/BF02762714
author: Jeffrey Bergen
author: Piotr Grzeszczuk
published in: Israel Journal of Mathematics
publication date: 1997-02-04T00:00:00Z
review text: Let \(R\) be a semiprime algebra over a field \(K\) acted on by a finite-dimensional nilpotent Lie superalgebra \(L\) (such that if \(\text{char }K=p\) then \(L\) is restricted, and if \(\text{char }K=0\) then \(L\) acts on \(R\) as algebraic derivations and algebraic superderivations). The purpose of this paper is to prove a series of ``going up'' results showing what the structure of the subalgebra of invariants \(R^L\) implies with respect to that of \(R\). In addition, these results raise a number of questions, and the authors give counterexamples to many of them. Some of the results in this paper are: If \(R^L\) is right Noetherian (Artinian), then \(R\) is a Noetherian (Artinian) right \(R^L\) module. If \(R^L\) is finite dimensional (has finite Goldie dimension as an \(R^L\)-module), then \(R\) is finite dimensional (has finite Goldie dimension as an \(R\)-module). If \(R\) is prime and \(R^L\) is central, then \(R\) satisfies a polynomial identity. The authors give examples to show that the reverse implications are not always true: for example they give an example of a prime left Noetherian ring \(R\) of arbitrary characteristic with a nilpotent derivation \(\delta\) such that \(R^{(\delta)}\) is not left Noetherian. In another example they show that certain hypotheses on \(R^L\) are needed in order to have \(R\) finitely generated over \(R^L\) even when \(R\) is prime and \(L\) is 1-dimensional.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 902631
zbMATH Keywords: semiprime algebras
zbMATH Keywords: finite-dimensional nilpotent Lie superalgebras
zbMATH Keywords: algebraic superderivations
zbMATH Keywords: subalgebra of invariants
zbMATH Keywords: finite Goldie dimension
zbMATH Keywords: polynomial identities
zbMATH Keywords: prime left Noetherian rings
zbMATH Keywords: nilpotent derivations
reviewed by: Davida Fischman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 902631