Auslander-Reiten conjecture for symmetric algebras of polynomial growth

TL;DR

This paper investigates the classification of certain self-injective algebras of polynomial growth, establishing equivalences between derived and stable classifications, and confirming the Auslander-Reiten conjecture for these cases.

Contribution

It proves the derived and stable equivalence classifications coincide for weakly symmetric algebras of polynomial growth, validating the Auslander-Reiten conjecture in this context.

Findings

Derived equivalence classification matches stable equivalence classification for domestic type.

Derived and stable classifications coincide for non-domestic polynomial growth algebras, up to scalar issues.

Auslander-Reiten conjecture holds for stable equivalences of Morita type between these algebras.

Abstract

This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for 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 of Morita type between weakly symmetric algebras of polynomial growth.