Local structure of tame symmetric algebras of period four

TL;DR

This paper characterizes the local structure of tame symmetric algebras of period four by showing their Gabriel quivers match those of weighted surface algebras, extending known classifications.

Contribution

It provides a detailed description of the Gabriel quivers for a specific class of tame symmetric algebras, linking them to weighted surface algebras.

Findings

Gabriel quivers are biregular with at most 2 arrows per vertex.

Local structure matches that of weighted surface algebras.

Extends classification of algebras of generalized quaternion type.

Abstract

In this paper we study the structure of Gabriel quivers of tame symmetric algebras of period four. More precisely, we focus on algebras having Gabriel quiver {\it biregular}, i.e. the numbers of arrows starting and ending at any vertex are equal, and do not exceed $2$. We describe the local structure of biregular Gabriel quivers of tame symmetric algebras of period four, including certain idempotent algebras. The main result of this paper shows that, in fact, these Gabriel quivers have local structure exactly as Gabriel quivers of so called {\it weighted surface algebras}, which partially extends known characterization of algebras of generalized quaternion type.