Hochschild cohomology of algebras of dihedral type, I: The family $D(3\mathcal K)$ in characteristic $2$

Abstract

In terms of generators and defining relations, the Hochschild cohomology algebras are described for all algebras of dihedral type in the family $D(3\mathcal {K})$ over an algebraically closed field of characteristic $2$. The results are applied to three other families of algebras of dihedral type, namely, $D(3\mathcal {A})_1$, $D(3\mathcal {B})_1$, and $D(3\mathcal {D})_1$. As a corollary, a description is obtained for the Hochschild cohomology algebra for blocks with dihedral defect group and three simple modules; in particular, this applies to principal blocks of the groups $\operatorname {PSL}(2,q)$ with odd $q$.