On a derived equivalence classification of weakly symmetric algebras of Euclidean type

Abstract

We consider a family $({\Lambda }_{\lambda }{)}_{\lambda \in {\mathbb{k}}^{\times }}$ of weakly symmetric algebras of Euclidean type over an algebraically closed field $\mathbb{k}$, where ${\mathbb{k}}^{\times }:=\mathbb{k}\setminus \{0\}$, and for each $\lambda \in {\mathbb{k}}^{\times }$, ${\Lambda }_{\lambda }$ is socle equivalent to ${\Lambda }_1$, which is the trivial extension of a hereditary algebra of type $\tilde{A}$. We compute their Hochschild cohomologies, which shows that ${\Lambda }_{\lambda }$ is not derived equivalent to ${\Lambda }_1$ if $\lambda \ne 1$. This means that a derived equivalence classification of weakly symmetric algebras of Euclidean type given by Bocian--Holm--Skowro{\' n}ski is not complete.

arXiv comment22 pages