Auslander–Reiten conjecture for symmetric algebras of polynomial growth

Abstract

This paper studies self-injective algebras of polynomial growth. We prove that two indecomposable weakly symmetric algebras of domestic type are derived equivalent if and only if they are stably equivalent. Furthermore we prove that for indecomposable weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander–Reiten conjecture for stable equivalences between weakly symmetric algebras of domestic type and for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth.