Realising higher cluster categories of Dynkin type as stable module categories

TL;DR

This paper demonstrates that stable module categories of specific selfinjective algebras of finite representation type are equivalent to u-cluster categories of the same Dynkin type, linking algebraic and categorical structures.

Contribution

It establishes a triangulated equivalence between stable module categories and u-cluster categories for algebras of Dynkin types, extending the understanding of their categorical relationships.

Findings

Stable module categories are triangulated equivalent to u-cluster categories.

The proof uses Keller and Reiten's Morita theorem and Dugas's Calabi-Yau dimension computations.

The result applies to algebras with tree class A_n, D_n, E_6, E_7, E_8.

Abstract

We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class A_n, D_n, E_6, E_7 or E_8 are triangulated equivalent to u-cluster categories of the corresponding Dynkin type. The proof relies on the 'Morita' theorem for u-cluster categories by Keller and Reiten, along with the recent computation of Calabi-Yau dimensions of stable module categories by Dugas.