Symmetric Hochschild cohomology of twisted group algebras

TL;DR

This paper introduces a new symmetric Hochschild cohomology theory for twisted group algebras by defining an action of the symmetric group on their Hochschild cochain complexes, extending previous symmetric cohomology concepts.

Contribution

It establishes the symmetric Hochschild cohomology for twisted group algebras and provides explicit embeddings and connecting homomorphisms between cohomology spaces.

Findings

Defined symmetric Hochschild cohomology for twisted group algebras

Constructed explicit embeddings between cohomology spaces

Established connecting homomorphisms for the cohomology

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 th construction of symmetric group cohomology due to Staic. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.