Derived equivalence of symmetric special biserial algebras

Abstract

We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial algebras, tilting complex.