Classification of symmetric special biserial algebras with at most one non-uniserial indecomposable projective

TL;DR

This paper classifies a specific class of symmetric special biserial algebras with limited non-uniserial projectives, providing explicit descriptions and equivalence classifications, extending previous work on related algebra types.

Contribution

It explicitly describes these algebras via quivers and relations and classifies them up to derived and stable Morita equivalences, expanding understanding of their structure.

Findings

Explicit quiver and relation descriptions of the algebras

Classification up to derived equivalence

Classification up to stable Morita equivalence

Abstract

We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the algebras of [Bocian-Holm-Skowro\'nski, J. Pure Appl. Algebra 2004], where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.