Ideals and quotients in crossed products of Hopf algebras
type: Item
instance of: scholarly article
title: Ideals and quotients in crossed products of Hopf algebras
zbMATH Open document ID: 0773.16019
DOI: 10.1016/0021-8693(92)90037-M
author: Jeffrey Bergen
author: Susan Montgomery
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: This paper is concerned with the relationships between ideals of crossed products \(R\#_ \sigma H\) and ideals of \(R\), and with certain quotient rings of these algebras, where \(H\) is a Hopf algebra over a field \(k\) and \(R\) is an \(H\)-module algebra. The main Hopf algebras under concern here are of the form \(K\# kG\), where \(K\) is irreducible and \(G\) is a group. Pointed cocommutative Hopf algebras are of this form. In Theorem 2.5: Essentially outer actions of such Hopf algebras on prime algebras \(R\) lead to a particularly nice connection: every non-zero ideal of \(R\#_ \sigma H\) intersects \(R\) non-trivially. These conditions lead in turn to interesting connections between the quotient rings: (1) \(C(R\#_ \sigma H)=C(R)^ H\), where \(C\) denotes the extended center of the appropriate quotient ring. (2) \(Q(R\#_ \sigma H)=Q_ H(R)\#_ \sigma H\), under an additional condition on regular elements. More is said when \(H\) is irreducible and cocommutative and \(\sigma\) is trivial.
reviewed by: Miriam Cohen
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16S40
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 130491
zbMATH Keywords: ideals of crossed products
zbMATH Keywords: \(H\)-module algebra
zbMATH Keywords: cocommutative Hopf algebras
zbMATH Keywords: outer actions
zbMATH Keywords: prime algebras
zbMATH Keywords: quotient rings
zbMATH Keywords: extended center
zbMATH Keywords: regular elements
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 130491