Invariants of the stable equivalence of symmetric special biserial algebras

Abstract

The present paper is one in a series of papers devoted to the classification of some classes of tame algebras up to stable category equivalence. In this paper, we study symmetric algebras (their stable categories have a structure of triangulated categories) and the simplest class of tame algebras-the class of special biserial algebras (SB-algebras). In the paper, we give a relevant version of the “diagrammatic method” and study the structure of the triangulated category “in a neighborhood” of the periodic part (with respect to Ω) of the stable category. Thus we prove the invariance of the collection of lengths of G-cycles under equivalence of stable categories (see Theorem 2.12). Then we use the invariance stated above, together with some properties of the Cartan matrix of a symmetric SB-algebra, to prove that the number of A-cycles (but not their lengths!) is also an invariant of stable equivalence. Bibliography: 8 titles.