The Grothendieck groups and stable equivalences of mesh algebras

TL;DR

This paper studies the Grothendieck groups of stable categories of finite-dimensional mesh algebras derived from stable translation quivers, providing a classification of these algebras up to stable equivalence.

Contribution

It determines the Grothendieck groups of stable categories of mesh algebras and classifies these algebras up to stable equivalence, a novel comprehensive analysis.

Findings

Grothendieck groups of stable categories are explicitly determined.

Complete classification of mesh algebras up to stable equivalence.

Stable categories have a triangulated structure due to self-injectivity.

Abstract

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable categories. As an application, we give an complete classification of the mesh algebras up to stable equivalences.