Cocharacter sequences of algebras with Hopf algebra actions
type: Item
instance of: scholarly article
title: Cocharacter sequences of algebras with Hopf algebra actions
zbMATH Open document ID: 0864.16019
author: Allan Berele
published in: Journal of Algebra
publication date: 1997-06-12T00:00:00Z
review text: The author develops the theory of cocharacters of polynomial identities of an algebra \(A\) with action of a finite dimensional Hopf algebra \(H\) over a field of characteristic 0. He proves analogues of the basic quantitative theorems for PI-algebra cocharacters. The results are in the spirit of the ordinary polynomial identities, *-polynomial identities for algebras with involution * and \(\mathbb{Z}/2\mathbb{Z}\)-graded polynomial identities. First the author defines the \(S_n\)- and the \(\text{GL}(k)\)-cocharacter sequences for the \(H\)-identities of \(A\) and the Poincaré (or Hilbert) series of the relatively free algebra with \(H\)-action. He establishes that these three invariants are equivalent. He compares them with their origins from the ordinary PI-theory and proves various bounds on the cocharacters. In particular, several quantitative characterizations of the property that an algebra with an \(H\)-identity is PI are given. Then the author introduces refinements of the above invariants involving representations of wreath products of \(H\). He extends his main theorems to this case when \(H\) is semisimple as an algebra and the field is algebraically closed. It turns out that the methods developed in the paper work in a more general setup. As a byproduct the author establishes some new results of independent interest concerning *-polynomial identities.
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 954786
zbMATH Keywords: cocharacter sequences of \(T\)-ideals
zbMATH Keywords: actions of finite dimensional Hopf algebras
zbMATH Keywords: graded polynomial identities
zbMATH Keywords: Poincaré series
zbMATH Keywords: Hilbert series
zbMATH Keywords: cocharacters of polynomial identities
zbMATH Keywords: PI-algebra cocharacters
zbMATH Keywords: *-polynomial identities
zbMATH Keywords: relatively free algebras
zbMATH Keywords: invariants
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
full work available at URL: jabr.1996.0354
OpenAlex ID: W2053547148
DOI: 10.1006/JABR.1996.0354
description: scientific article; zbMATH DE number 954786