The ubiquity of generalized cluster categories

TL;DR

This paper demonstrates that many 2-Calabi-Yau triangulated categories, including those linked to Coxeter group elements, are equivalent to generalized cluster categories, broadening the understanding of their structure.

Contribution

It proves that a wide class of 2-Calabi-Yau categories are triangle equivalent to generalized cluster categories, extending previous results to more cases.

Findings

Many 2-Calabi-Yau categories are equivalent to generalized cluster categories.

Includes categories associated with Coxeter group elements.

Generalizes earlier special case results.

Abstract

Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when it is $\Hom$-finite. By definition, the cluster categories of \cite{Bua} are a special case. In this paper we show that a large class of 2-Calabi-Yau triangulated categories, including those associated with elements in Coxeter groups from \cite{Bua2}, are triangle equivalent to generalized cluster categories. This was already shown for some special elements in \cite{Ami3}.