Hochschild cohomology for algebras of semidihedral type. X: Cohomology algebra for the exceptional local algebras (Q6043872)

The paper under review is concerned with a family of finite-dimensional local algebras \(R = R_{k,c,d}\) defined over an algebraically closed field \(K\) of characteristic \(2\); here \(k \geq 2\) is an integer, and \(c,d\) are elements of \(K\) such that \((c,d) \neq (0,0)\). These algebras are tame and symmetric and generated by two elements, and they appear on \textit{K. Erdmann}'s list of algebras related to tame blocks of finite groups [Blocks of tame representation type and related algebras. Berlin etc.: Springer-Verlag (1990; Zbl 0696.20001)]. The main result of the paper describes the structure of the Hochschild cohomology algebra \(\mathrm{HH}^\ast(R)\), as a graded algebra, via generators and relations. There are six cases, depending on properties of the parameters \(k,c,d\). In case (1), \(\mathrm{HH}^\ast(R)\) has \(13\) homogeneous generators and \(25\) types of relations; the other cases are similar. For Part IX see [the author and \textit{D. A. Nikulin}, J. Math. Sci., New York 247, No. 4, 507--517 (2020; Zbl 1468.16016); translation from Zap. Nauchn. Semin. POMI 478, 17--31 (2019)].