On the first Hochschild cohomology group of a cluster-tilted algebra

ASSEM, IBRAHIM; REDONDO, MARIA JULIA; SCHIFFLER, RALF

Algebras and Representation Theory

SPRINGER

Lugar: Berlin; Año: 2015 vol. 18 p. 1547 - 1576

1386-923X

Given a cluster-tilted algebra $B$, we study its first Hochschild cohomology group ${HH}^1(B)$ 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 ${HH}^1(B)$ is isomorphic, as a $k$-vector space, to the direct sum of ${HH}^1(C)$ with $k^{n_{B,C}}$, where $n_{B,C}$ is an invariant linking the bound quivers of $B$ and $C$. In the representation-finite case, ${HH}^1(B)$ can be read off simply by looking at the quiver of $B$.