Classifying tame blocks and related algebras up to stable equivalences of Morita type

TL;DR

This paper classifies certain finite-dimensional algebras of dihedral, semi-dihedral, and quaternion type under stable equivalence of Morita type, confirming the Auslander-Reiten conjecture for these classes.

Contribution

It provides a classification of Erdmann's algebras of specific types and verifies the Auslander-Reiten conjecture for them, advancing the understanding of algebra classification.

Findings

Classification of Erdmann's algebras of dihedral, semi-dihedral, and quaternion type.

Validation of the Auslander-Reiten conjecture for these classes.

Enhanced understanding of stable equivalences of Morita type.

Abstract

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of the class of Erdmann's algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for these classes of algebras.