Cocommutative Hopf algebra actions and the Connes spectrum
type: Item
instance of: scholarly article
title: Cocommutative Hopf algebra actions and the Connes spectrum
zbMATH Open document ID: 0809.16034
author: James Osterburg
author: Declan Quinn
published in: Journal of Algebra
publication date: 1995-04-02T00:00:00Z
review text: The Connes spectrum, as introduced [in Contemp. Math. 130, 311-334 (1992; Zbl 0782.16021)] by the authors and the reviewer, is too technically to formally define here. Roughly speaking, if \(H\) is a finite-dimensional, semisimple Hopf algebra over a splitting field \(K\), then \(\text{CS}(A,H)\) is a subset of the irreducible representations of \(H\) determined by the action of \(H\) on the \(H\)-module algebra \(A\). Indeed, the Connes spectrum \(\text{CS}(A,H)\) is intimately related to the smash product \(A \# H\) and can be used to study the simplicity and primeness of the latter ring. This paper proves a lovely, rather surprising result, namely if \(H\) is cocommutative and if \(A\) is \(H\)-prime, then \(\text{CS}(A,H)\) is closed under multiplication. In other words, if \(\pi,\chi \in\text{CS}(A,H)\), then all irreducible constituents of the tensor representation \(\pi \otimes \chi\) are also contained in the Connes spectrum. As a consequence, in the case of group algebras, it follows that \(\text{CS}(A,K[G])\) consists precisely of all the irreducible \(K[G]\)- modules which factor through \(K[G/N]\) for some fixed normal subgroup \(N\) of \(G\). The paper then goes on to study the relationship between this Connes kernel \(N=\text{CK}(A,K[G])\) and the action of \(G\) on \(A\).
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 589953
zbMATH Keywords: Connes spectrum
zbMATH Keywords: finite-dimensional, semisimple Hopf algebra
zbMATH Keywords: splitting field
zbMATH Keywords: irreducible representations
zbMATH Keywords: action
zbMATH Keywords: \(H\)-module algebra
zbMATH Keywords: smash product
zbMATH Keywords: simplicity
zbMATH Keywords: primeness
zbMATH Keywords: tensor representation
zbMATH Keywords: group algebras
reviewed by: D. S. Passman
MaRDI profile type: Publication
full work available at URL: jabr.1994.1124
OpenAlex ID: W2013309948
DOI: 10.1006/JABR.1994.1124
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description: scientific article; zbMATH DE number 589953