Symmetric Hochschild cohomology of twisted group algebras

Abstract

We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly to Staic's construction of symmetric group cohomology. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.

Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras

“…We say that A is cocommutative if tw • ∆ = ∆, where tw : A ⊗ A − A ⊗ A is the morphism given by tw(a ⊗ b) = b ⊗ a for any a, b ∈ A. We will use some standard notation for the coproduct, so called Sweedler notation; we write ∆(a) = a (1) ⊗ a (2) , where the notation a (1) , a (2) for tensor factors is symbolic. Throughout the paper, we omit the summation symbol of Sweedler notation when no confusion occurs.…”

“…(1) The k-vector space M ⊗ N is a left A-module via a • (m ⊗ n) = a (1) m ⊗ a (2) n for any a ∈ A, m ∈ M and n ∈ N.…”

“…(2) The k-vector space Hom k (M, N) is a left A-module via (a • f )(m) = a (1) f (S(a (2) )m) for any a ∈ A, m ∈ M and f ∈ Hom k (M, N). In particular, if N = k, then Hom k (M, k) has a left A-module structure given by (a…”

“…(4) Let M be an A-bimodule. The left action on M is defined by a • m = a (1) mS(a (2) ) for any a ∈ A and m ∈ M, which is called the left adjoint action. Also, let ad M denote by a left A-module M with the left adjoint action.…”

“…Singh [6] defined the symmetric continuous cohomology of topological groups and the symmetric smooth cohomology of Lie groups. Recently, Coconet-Todea [2] defined the symmetric Hochschild cohomology of twisted group algebras which is a generalization of group algebras. Our aim of this paper is to study about the symmetric cohomology and the symmetric Hochschild cohomology for cocommutative Hopf algebras as another generalization of group algebras.…”