Hochschild cohomology of relation extension algebras

IBRAHIM ASSEM, MARÍA ANDREA GATICA Y RALF SCHIFFLER; MARIA ANDREA GATICA; RALF SCHIFFLER; RACHEL THAILLEFER

Mar del Plata

Congreso; ARTA (Advances in Representation Theory of Algebras); 2016

Let $B$ be the split extension of a finite dimensional algebra $C$ by a $C-C$-bimodule $E$. The objective of this talk is to describe a morphism of associative graded algebras $\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$ and give necessary and sufficient conditions for the surjectivity of $\varphi^*$ in the case that $B$ is a trivial extension of $C$ by $E$. In particular, when $B$ is a relation extension of a triangular algebra $C$ of global dimension at most two by the $C-C$-bimodule $E=\Ext^2_C(C,DC)$ we describe the kernel of $\varphi^0$ and $\varphi^1$ in homological terms.