Finite mutation classes of coloured quivers

TL;DR

This paper characterizes when the mutation class of a coloured quiver, linked to m-cluster tilting objects, is finite, extending known results from 1-cluster categories to more general settings.

Contribution

It generalizes the classification of finite mutation classes of coloured quivers to broader algebraic contexts involving m-cluster tilting objects.

Findings

Mutation class is finite iff H is of finite or tame type, or has at most 2 simples.

Extends known results from 1-cluster categories.

Provides criteria for finiteness of mutation classes in coloured quivers.

Abstract

We consider the general notion of coloured quiver mutation and show that the mutation class of a coloured quiver $Q$, arising from an $m$-cluster tilting object associated with $H$, is finite if and only if $H$ is of finite or tame representation type, or it has at most 2 simples. This generalizes a result known for 1-cluster categories.