Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

The first Hochschild cohomology group of a schurian cluster-tilted algebra (Joint work with Ibrahim Assem)

Cluster-tilted algebras were defined by Buan, Marsh and Reiten, and have been studied by several authors. Our objective is to study the first Hochschild cohomology group ${\rm HH}^1(B)$ of a cluster-tilted algebra $B$. As a first step, we consider the case where $B$ is schurian: this includes the case of all representation-finite cluster-tilted algebras. Under this assumption we have the following result: if $C$ is a tilted algebra, then the trivial extension of $C$ by the $C$-$C$-bimodule ${\rm Ext}^2_C(DC,C)$ is cluster-tilted, and, conversely, any cluster-tilted algebra is of this form. As a consequence, one can describe the ordinary quiver of $B$ knowing that of $C$. The main step in our proof consists in defining an equivalence relation between the arrows in the quiver of $B$ which are not in the quiver of $C$. We get a short exact sequence connecting ${\rm HH}^1(C)$, ${\rm HH}^1(B)$ and the $k$-vector space of dimension $n_{B,C}$, the number of equivalence classes.