Hochschild cohomology ring of a cluster-tilted algebra of type D4

Abstract

In this paper we describe the Hochschild cohomology rings for algebras in a class of some special biserial algebras which contains a cluster-tilted algebra of Dynkin type ${\mathbb{D}}_4$. In particular it is shown that the Hochschild cohomology rings modulo nilpotence for these algebras are isomorphic to the polynomial ring $K[x]$. As an application we prove that the cluster-tilted algebra of type ${\mathbb{D}}_4$ contained in this class satisfies the finiteness conditions (Fg1) and (Fg2) introduced in [EHSST].