On the First Hochschild Cohomology Group of a Cluster-Tilted Algebra

Abstract

Given a cluster-tilted algebra B, we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation-extension of C, then we show that if B is tame, then HH1(B) is isomorphic, as a k-vector space, to the direct sum of \({\text {HH}}^{1}(C)\) with \(k^{n_{B,C}}\), where nB,C is an invariant linking the bound quivers of B and C. In the representation-finite case, HH1(B) can be read off simply by looking at the quiver of B.