Hochschild cohomology of relation extension algebras

Ibrahim Assem (1) , M. Andrea Gatica (2) , Ralf Schiffler (3) , Rachel Taillefer (4)

1 UdeS - Département de mathématiques [Sherbrooke]
2 UNS - Universidad Nacional del Sur [Argentina]
3 UCONN - University of Connecticut
4 LMBP - Laboratoire de Mathématiques Blaise Pascal

Let $B$ be the split extension of a finite dimensional algebra $C$ by a $C$-$C$-bimodule $E$. We define a morphism of associative graded algebras $\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension $B=C\ltimes E$, we give necessary and sufficient conditions for each $\varphi^n$ to be surjective. We prove the surjectivity of $\varphi^1$ for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of $\varphi^1$ for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.

Mots clés: Hochschild cohomology; trivial extensions; relation extensions

Domaines: Théorie des représentations [math.RT]

Ibrahim Assem, M. Andrea Gatica, Ralf Schiffler, Rachel Taillefer. Hochschild cohomology of relation extension algebras. 2015. ⟨hal-01179207v1⟩