Algebras of generalized quaternion type: biregular case

TL;DR

This paper advances the classification of generalized quaternion type algebras by analyzing biregular cases, showing they are either weighted surface algebras or higher spherical algebras, thus extending previous classifications.

Contribution

It classifies algebras with biregular Gabriel quivers, expanding the understanding of their structure and types within the generalized quaternion framework.

Findings

Algebras are either weighted surface or higher spherical algebras.

Classification extends previous work on 2-regular Gabriel quivers.

Main result applies up to socle equivalence.

Abstract

This paper provides the next step towards classification of algebras of generalized quaternion type. Previously algebras with 2-regular Gabriel quiver were classified (a quiver is 2-regular if at each vertex, two arrows start and two arrows end). Here we classify the algebras where at each vertex, either one arrow starts and one arrow ends, or else two arrows start and two arrows end. Our main result shows that that any such algebra (up to socle equivalence) is either a weighted surface algebra, or a higher spherical algebra.