# The Batalin-Vilkovisky structure on the Tate-Hochschild cohomology ring of a group algebra

## TL;DR

This paper characterizes the Batalin-Vilkovisky structure on the Tate-Hochschild cohomology of a group algebra, revealing its subalgebra relations with Tate cohomology and cyclic $A_{}$-structures.

## Contribution

It explicitly describes the Batalin-Vilkovisky structure on Tate-Hochschild cohomology for group algebras and establishes subalgebra and cyclic $A_{}$-relations.

## Key findings

- Tate cohomology forms a BV subalgebra of Tate-Hochschild cohomology.
- The Tate cochain complex is a cyclic $A_{}$-subalgebra of the Hochschild cochain complex.
- Explicit decomposition of the cohomology ring structure.

## Abstract

We determine the Batalin-Vilkovisky structure on the Tate-Hochschild cohomology of the group algebra $kG$ of a finite group $G$ in terms of the additive decomposition. In particular, we show that the Tate cohomology of $G$ is a Batalin-Vilkovisky subalgebra of the Tate-Hochschild cohomology of the group algebra $kG$, and that the Tate cochain complex of $G$ is a cyclic $A_{\infty}$-subalgebra of the Tate-Hochschild cochain complex of $kG$.