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

Abstract

We consider a family $(Λ_λ)_{λ\in \Bbbk^\times}$ of weakly symmetric algebras of Euclidean type over an algebraically closed field $\Bbbk$, where $\Bbbk^\times:= \Bbbk\setminus \{0\}$, and for each $λ\in \Bbbk^\times$, $Λ_λ$ is socle equivalent to $Λ_1$, which is the trivial extension of a hereditary algebra of type $\tilde{A}$. We compute their Hochschild cohomologies, which shows that $Λ_λ$ is not derived equivalent to $Λ_1$ if $λ\neq 1$. This means that a derived equivalence classification of weakly symmetric algebras of Euclidean type given by Bocian--Holm--Skowro{\' n}ski is not complete.