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 connections 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 the known class of weighted surface algebras. The second provides new classifications of TSP4 algebras (based on known results for 2-regular case), which handle algebras for which Gabriel quivers satisfy more general properties; the classification in biregular case is known (the biserial case is a current work in progress). An ongoing project sheds a new light on the combinatorics of such algebras, introducing a useful new tool for their classification called periodicity shadows. In this paper, we address the problem of classifying TSP4 algebras from another perspective: we provide a classification of all TSP4 algebras with not-too-big Gabriel quivers, i.e., having at most 5 vertices, but without restrictions on their structure, which differentiates it from previous classifications. The result is based on the application of the concept of the periodicity shadow, which allows all possible Gabriel quivers of such algebras to be computed (for small number of vertices) as well as on recent results concerning iterated mutations of algebras with periodic simple modules. The main result shows that TSP4 algebras with at most 5 vertices are generalized weighted surface algebras, confirming a general conjecture in this case.