Tame symmetric algebras of period four with small Gabriel quivers

Abstract

The tame symmetric algebras of period four, TSP4 algebras for short, form an important class of algebras, with interesting links to various branches of modern algebra. The study of this class has been recently developed in two major directions. The first embraces new classes of examples of TSP4 algebras, such as virtual mutations and generalized weighted surface algebras, both extending known class of the weighted surface algebras. The second provides new classifications of TSP4 algebras (based on known results for $2$-regular case), which handle algebras, whose Gabriel quivers satisfy more general properties. An ongoing project shades a new light on the combinatorics of such algebras, introducing a new useful tool for their classification, called periodicity shadows. In this paper, we attack the problem of classification of TSP4 algebras, from another perspective, namely, we give a classification of all TSP4 algebras with not too big Gabriel quivers, i.e. having at most $5$ vertices -- but with no restrictions on their structure, as it was the case for previous classifications. The result is based on the application of the notion of periodicity shadow, which allows to compute all possible Gabriel quivers of such algebras (for small number of vertices), and recent results on interated mutations of algebras with periodic simple modules. The main result show that TSP4 algebras with Gabriel quivers having at most $5$ vertices are generalized weighted surface algebras, confirming a general conjecture in this case.