$BV$-structure on Hochschild cohomology for exceptional local algebras of quaternion type. The case of an even parameter

A full description is provided for the $BV$-structure on the Hochschild cohomology of exceptional local algebras of quaternion type determined by parameters $(k,0,d)$ in the case of an even parameter $k \geqslant 3$, according to Erdmann's classification. The method of comparison morphisms and the weak self-homotopy method are developed and used. This article states as a generalization of similar results about the $BV$-structures on Hochschild cohomology of algebras of quaternion type.