Hochschild Cohomology Ring Of A Cluster-tilted Algebra

Hochschild cohomology theory which was introduced by Hochschild and devel-oped by Cartan and Eilenberg is a component of homological algebra, it plays an im-portant role in the representation theory of algebra. The Hochschild cohomology space of finite dimensional algebra is a graded commutative ring under the cup product. In this paper we study the Hochschild cohomology ring of a special cluster-tilted algebra. The algebra is a special biserial Koszul algebra, but not selfinjective algebra. We first give an explicit description of the so-called "comultiplicative structure" based on the minimal projective bimodule resolution constructed by Furuya, and thus show that the cup product in the level of cochains for the special cluster-tilted algebra is essentially juxtaposition of parallel paths up to sign. As a consequence, we determine the struc-ture of the Hochschild cohomology ring under the cup product by giving an explicit presentation by generators and relations.