On the Hochschild cohomology ring of the quaternion group of order eight in characteristic two

TL;DR

This paper computes the detailed algebraic structures, including Gerstenhaber and Batalin-Vilkovisky structures, on the Hochschild cohomology ring of the quaternion group of order eight over a field of characteristic two.

Contribution

It provides the first explicit determination of the Gerstenhaber and Batalin-Vilkovisky structures for this specific group algebra.

Findings

Gerstenhaber Lie algebra structure explicitly determined

Batalin-Vilkovisky structure explicitly computed

Hochschild cohomology ring structure clarified in characteristic two

Abstract

Let $k$ be an algebraically closed field of characteristic two and let $Q_8$ be the quaternion group of order $8$. We determine the Gerstenhaber Lie algebra structure and the Batalin-Vilkovisky structure on the Hochschild cohomology ring of the group algebra $kQ_8$.