Invariant algebras and completely reducible representations
type: Item
instance of: scholarly article
title: Invariant algebras and completely reducible representations
zbMATH Open document ID: 0849.16036
author: Hanspeter Kraft
author: Lance W. Small
published in: Mathematical Research Letters
publication date: 1996-11-03T00:00:00Z
review text: Assume that \(K\) is a field. The authors present a construction of a large class of affine noetherian \(K\)-algebras \(A\) satisfying the following condition: (FCR) Every finite dimensional representation of \(A\) is completely reducible and the intersection of the kernels of all finite dimensional representations of \(A\) is zero. In particular it is shown that if \(B\) is a noetherian algebra satisfying the condition (BCR) and \(H\) is a finite group of automorphisms of \(B\) such that \((|H|,\text{char}(K))=1\) then the algebra \(B^H\) of \(H\)-invariants of \(B\) also satisfies (BCR). If \(K\) is algebraically closed of characteristic zero, \(\mathfrak g\) is a semisimple Lie \(K\)-algebra, \(B={\mathfrak U}({\mathfrak g})\) is the universal enveloping algebra of \(\mathfrak g\) and \(H\) is a finite group of automorphisms of \(\mathfrak g\) then \(B\) is affine, prime, left and right noetherian and the algebras \(B\), \(B^H\) satisfy the condition (FCR). It is also proved that a \(K\)-algebra \(A\) satisfies (BCR) if and only if \(\bigcap_{m\geq 0}I_m(A)=(0)\) and the factor algebra \(A/I_m(A)\) is finite dimensional and semisimple for every \(m\geq 0\), where \(I_m(A)\subseteq A\) is the ideal of \(A\) of polynomial identities of degree \(m\).
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 817696
zbMATH Keywords: affine Noetherian algebras
zbMATH Keywords: finite dimensional representations
zbMATH Keywords: finite group of automorphisms
zbMATH Keywords: universal enveloping algebras
zbMATH Keywords: polynomial identities
MaRDI profile type: Publication
DOI: 10.4310/MRL.1994.V1.N3.A2
description: scientific article; zbMATH DE number 817696