# A BV-algebra Structure on Hochschild Cohomology of the Group Ring of Finitely Generated Abelian Groups

## TL;DR

This paper investigates the Batalin-Vilkovisky algebra structure on the Hochschild cohomology of group rings of finitely generated abelian groups, connecting algebraic structures to properties like symmetry and Calabi-Yau conditions.

## Contribution

It establishes a BV-algebra framework for Hochschild cohomology of these group rings, extending known results from finite and free abelian groups.

## Key findings

- BV-algebra structure exists for finite abelian groups due to symmetry.
- For free abelian groups, the structure arises from Calabi-Yau properties.
- Provides a unified approach to Hochschild cohomology in this context.

## Abstract

We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.