Algebras of generalized quaternion type

TL;DR

This paper introduces and analyzes algebras of generalized quaternion type, showing they are periodic of period 4, related to surface algebras, and contributes to classifying periodic symmetric tame algebras.

Contribution

It establishes the periodicity and deformation structure of generalized quaternion type algebras, advancing the classification of periodic symmetric tame algebras.

Findings

All such algebras with 2-regular Gabriel quivers are periodic of period 4.

They are specific deformations of weighted surface algebras.

Existence of wild periodic algebras with many simple modules.

Abstract

We introduce and study the algebras of generalized quaternion type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these algebras, with 2-regular Gabriel quivers, are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles. The main result o f the paper forms an important step towards the Morita equivalence classification of all periodic symmetric tame algebras of non-polynomial growth. Applying the main result, we establish existence of wild periodic algebras of period 4, with arbitrarily large number (at least 4) of pairwise non-isomorphic simple modules. These wild periodic algebras arise as stable endomorphism rings of cluster tilting Cohen-Macaulay modules over one-dimensional hypersurface singularities.