The first Hochschild cohomology group of a schurian cluster-tilted algebra

Abstract

AbstractGiven a cluster-tilted algebra B we study its first Hochschild coho-mology group HH 1 (B) with coefficients in the B-B-bimodule B. We findseveral consequences when B is representation-finite, and also in the casewhere B is cluster-tilted of type A˜.2000 Mathematics Subject Classification : 16E40 1 Introduction Cluster categories were introduced in [11] and also in [17] for type A, in order tounderstand better the cluster algebras of Fomin and Zelevinsky [20]. Cluster-tiltedalgebras were defined in [12] and also in [18] for type A. These algebras have beenstudied by several authors (see, for instance, [1, 18, 12, 13]). Our objective here is,for a cluster-tilted algebra B, to study its first Hochschild cohomology group HH 1 (B)with coefficients in the B-B-bimodule B, see [19]. As a first step, we consider thecase where B is schurian: this includes the case of all representation-finite cluster-tiltedalgebras. There are several reasons for this restriction. Indeed, it was shown in [1]that, if C is a tilted algebra, then the trivial extension C ⋉ Ext