Crossed products, cohomology, and equivariant projective representations
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 \(H^ 2(G,K^*)\).
scientific article; zbMATH DE number 4107040