The Batalin-Vilkovisky Structure on the Tate-Hochschild Cohomology Ring of a Group Algebra

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 $kG$.