Irreducible representations of crossed products
type: Item
instance of: scholarly article
title: Irreducible representations of crossed products
zbMATH Open document ID: 0932.16039
DOI: 10.1016/S0022-4049(97)00077-7
author: Susan Montgomery
author: Sarah Witherspoon
published in: Journal of Pure and Applied Algebra
publication date: 1999-12-14T00:00:00Z
review text: In 1975, I. Kaplansky conjectured that if \(H\) is a finite-dimensional semisimple Hopf algebra over an algebraically closed field, then the dimension of each irreducible \(H\)-module divides the dimension of \(H\). The main result of this paper is a verification of this conjecture for fields of characteristic 0 and Hopf algebras of prime power dimension. The proof is based on a rather interesting general approach. Let \(k\) be an algebraically closed field and let \(H\) be a finite-dimensional Hopf algebra over \(k\) with the property that the dimension of each irreducible \(H\)-module divides the dimension of \(H\). If \(G\) is a finite group whose order is not divisible by the characteristic of \(k\), then it is shown here that any crossed product of the form \(H\#kG\) or \(H\#kG^*\) has this same property with respect to irreducible modules. In particular, this divisibility property is satisfied by Hopf algebras constructed as a sequence of crossed products involving group algebras and their duals. Finally, it is shown that if \(\text{char }k=0\), then any Hopf algebra of prime power dimension is of the above form, thereby yielding the main result.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 1308202
zbMATH Keywords: semisimple Hopf algebras
zbMATH Keywords: crossed products
zbMATH Keywords: irreducible representations
zbMATH Keywords: irreducible modules
zbMATH Keywords: finite-dimensional Hopf algebras
zbMATH Keywords: Hopf algebras of prime power dimension
reviewed by: D. S. Passman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308202