$BV$-algebra structure on Hochschild cohomology of local algebras of quaternion type in characteristic 2

This paper is a sequel of joint paper by the author with S. O. Ivanov, Yu. Volkov, and G. Zhou. In the present paper $BV$-structure, and therefore, Gerstenhaber algebra structure on Hochschild cohomology of local algebras of generalized quaternion type is completely described over a field of characteristic 2. The family of algebras under investigation contains group algebras of generalized quaternion groups for which the case of characteristic 2 is the only case when calculation of Hochschild cohomology and structures on it is a highly non-trivial problem. Also the group algebras of generalized quaternion groups represent classes of Morita-equivalence of tame group blocks from K. Erdmann's classification. So in particular $BV$-structure on Hochschild cohomology of group algebras of some non-commutative groups is described.