Crossed products, cohomology, and equivariant projective representations
type: Item
instance of: scholarly article
title: Crossed products, cohomology, and equivariant projective representations
zbMATH Open document ID: 0676.16023
DOI: 10.1016/0021-8693(89)90087-2
author: Eli Aljadeff
author: Shmuel Rosset
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: For a commutative algebra K over a commutative ring k and a homomorphism of groups t: \(G\to Aut_ kK\) the crossed product construction (CPC) [the authors, J. Pure Appl. Algebra 40, 103-113 (1986; Zbl 0584.16014)] associates to any \(\alpha \in H^ 2(G,K^*)\) and t a certain k-algebra \(K_ t^{\alpha}G\) which, considered together with a canonical map \(K\to K_ t^{\alpha}G\), is called here a K/k-algebra. Let \({\mathcal C}\) be a monoid of isomorphism classes of K/k-algebras with Sweedler multiplication \(\times_ K\) [\textit{M. E. Sweedler}, Publ. Math., Inst. Hautes Etud. Sci. 44, 79-189 (1974; Zbl 0314.16008)], \(H=\ker t\). The authors first show that if K is a domain and \(H=\{1\}\) then CPC: \(H^ 2(G,K^*)\to {\mathcal C}\) is injective and extend this result to the case of non-trivial H generalizing the Sweedler product so that for certain \(\alpha,\beta \in H^ 2(G,K^*)\) \(K_ t^{\alpha}G\times_ KK_ t^{\beta}G\sim K_ t^{\alpha \beta}G\). They introduce, secondly, a new type of representations of G on KG - equivariant projective representations (EPR) - and prove that up to a certain equivalence EPR are classified by the group \(H^ 1(G,Aut_ K(KG)/K^*)\) whereas the class group of EPR with diagonal matrix is isomorphic to \(H^ 2(G,K^*)\).
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 4107040
zbMATH Keywords: crossed product construction
zbMATH Keywords: monoid of isomorphism classes
zbMATH Keywords: Sweedler multiplication
zbMATH Keywords: equivariant projective representations
zbMATH Keywords: class group
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90087-2
OpenAlex ID: W2092047998
cites work: Global dimensions of crossed products
cites work: Q3856855
cites work: Groups of simple algebras
description: scientific article; zbMATH DE number 4107040