Quasi-biserial algebras, special quasi-biserial algebras and symmetric fractional Brauer graph algebras

TL;DR

This paper introduces quasi-biserial algebras as a new generalization of biserial algebras, establishing their properties, a combinatorial model via ribbon graphs, and derived equivalences through graph mutations.

Contribution

It defines quasi-biserial algebras, links symmetric cases to ribbon graphs with multiplicities, and shows how graph mutations induce derived equivalences.

Findings

Quasi-biserial algebras retain key properties of biserial algebras.

Symmetric special quasi-biserial algebras correspond to labeled ribbon graphs.

Mutations of ribbon graphs induce derived equivalences between algebras.

Abstract

Biserial algebras are a classical class in the representation theory of algebras, generalizing Nakayama algebras. They were further generalized by Green and Schroll to multiserial algebras, which share many structural properties with biserial algebras. Inspired by their motivation, we introduce another generalization, called quasi-biserial algebras. We show that this class retains fundamental properties of classical biserial algebras. In the symmetric special case, we establish a correspondence with labeled ribbon graphs equipped with multiplicities, providing a combinatorial model for the algebras. Furthermore, we prove that Kauer moves on these graphs, interpreted as mutations of labeled ribbon graphs, induce derived equivalences between the associated symmetric special quasi-biserial algebras.