Distinguishing derived equivalence classes using the second Hochschild cohomology group

TL;DR

This paper investigates the second Hochschild cohomology groups of specific algebras to distinguish their derived equivalence classes, providing explicit calculations and answering a previously posed question.

Contribution

It computes the second Hochschild cohomology for a preprojective algebra of type D4 and a related socle deformation, demonstrating they are not derived equivalent.

Findings

The second Hochschild cohomology groups differ for the two algebras.

The non-standard algebra is not derived equivalent to the preprojective algebra.

The work answers a question posed by Holm and Skowroński.

Abstract

In this paper we study the second Hochschild cohomology group of the preprojective algebra of type $D_4$ over an algebraically closed field $K$ of characteristic 2. We also calculate the second Hochschild cohomology group of a non-standard algebra which arises as a socle deformation of this preprojective algebra and so show that the two algebras are not derived equivalent. This answers a question raised by Holm and Skowro\'nski.