Hochschild Cohomology of Algebras of Semidihedral Type. I. Group Algebras of Semidihedral Groups

For a family of local algebras of semidihedral type over an algebraically closed field of characteristic 2, the Hochschild cohomology algebra is described in terms of generators and relations. The calculations are based on the construction of a bimodule resolution for the algebras in question. As a consequence, the Hochschild cohomology algebra is described for the group algebras of semidihedral groups. Introduction In recent years, interest has grown considerably in the investigation of the Hochschild cohomology algebra, and appreciable success has been made for the case of finite-dimensional algebras over fields. In [1], a description was obtained of the Hochschild cohomology algebra for the symmetric group S3 over the field F3, and also for the alternating group A4 and the dihedral 2-groups over the field F2. In [2], the Hochschild cohomology algebra HH∗(R) was described in the case where R is a self-injective Nakayama algebra. In [3], a description of the Hochschild cohomology algebra was given for algebras of dihedral type in the family D(3K) over an algebraically closed ground field of characteristic two. In that paper, unlike [1], we applied a direct method of constructing a bimodule resolution for the corresponding algebras, with subsequent use of this resolution in the calculation of Hochschild cohomology groups and multiplication in the cohomology algebra. We recall that algebras of dihedral, semidihedral, and quaternion types appeared in the work of K. Erdmann on the classification of group blocks of tame representation type (see [4]). Later, the approach of [3] was applied to several families of algebras of quaternion type. Namely, in [5], the Hochschild cohomology algebras were calculated for a family of local algebras of quaternion type, and in [6, 7], the algebra HH∗(R) was described for the family Q(2B)1 over an algebraically closed field of characteristic 2. We notice that bimodule resolutions for algebras of quaternion type were constructed also in [8]. Moreover, the approach of [3] was used for the calculation of the algebra HH∗(R) for Liu–Schulz algebras (see [9, 10]) and for the integer group ring Z[D4m] of the “even” dihedral group (i.e., of order 4m) [11]. Note that an independent description of the algebra HH(Z[D2m]) for all dihedral groups was obtained in [12], where the methods of the paper [1] were used. Some particular results were obtained in the description of the algebra HH∗(R) for the so-called Möbius algebra (see [13, 15]), for a family of self-injective algebras of finite representation type with tree class Dn (see [16]), and for group blocks of tame representation type that have one or three simple modules (see [17]). 2000 Mathematics Subject Classification. Primary 13D03.