Algebras of generalized quaternion type

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.