Hochschild cohomology of relation extension algebras

TL;DR

This paper investigates the Hochschild cohomology of relation extension algebras, defining a morphism between the cohomologies of the extension and the original algebra, and analyzing conditions for surjectivity and kernel structure.

Contribution

It introduces a morphism linking Hochschild cohomologies of split extensions and original algebras, providing criteria for surjectivity and detailed kernel descriptions in relation extensions.

Findings

Defined a morphism :HH^*(B) ddd:HH^*(C)

Established necessary and sufficient conditions for surjectivity of d^n in trivial extensions

Proved surjectivity of d^1 for classes including cluster-tilted algebras

Abstract

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.