Tame symmetric algebras of period four

Abstract

In this paper we are concerned with the structure of tame symmetric algebras of period four (TSP4 algebras, for short). We will mostly focus on the case when the Gabriel quiver of $A$ is biserial, i.e. there are at most two arrows ending and at most two arrows starting at each vertex, but some of the results can be easily extended to the general case. This serves as a basis for upcoming series of articles devoted to solve the problem of classification of all TSP4 algebras with biserial Gabriel quiver. We present a range of properties (with relatively short proofs) which must hold for the Gabriel quiver of a tame symmetric algebra of period four. Amongst others we show that triangles (and squares) appear naturally in the Gabriel quivers of such algebras, such as for weighted surface algebras [6, 8, 9].