$m$-cluster categories and $m$-replicated algebras

TL;DR

This paper establishes a correspondence between m-cluster categories of hereditary algebras and certain tilting modules over m-replicated algebras, providing a new structural insight into their relationship.

Contribution

It introduces the m-left part of m-replicated algebras as a fundamental domain for m-cluster categories and links tilting objects across these structures.

Findings

Identifies the m-left part as a fundamental domain for m-cluster categories.

Establishes a bijection between m-clusters and specific tilting modules.

Provides a structural framework connecting m-cluster categories and m-replicated algebras.

Abstract

Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category of A is the m-left part of the m-replicated algebra $A^{(m)}$ of A. Moreover, we obtain a one-to-one correspondence between the tilting objects in the m-cluster category (that is, the m-clusters) and those tilting $A^{(m)}$-modules for which all non projective-injective direct summands lie in the m-left part of $A^{(m)}$.