Relative cluster categories and Higgs categories

TL;DR

This paper extends the theory of cluster categories to a relative setting, introducing Higgs categories that are stably n-Calabi-Yau and Hom-finite, with applications to various algebraic structures.

Contribution

It generalizes higher cluster categories to the relative context, proving the existence of n-cluster tilting objects in Higgs categories derived from Frobenius extriangulated categories.

Findings

Higgs categories are stably n-Calabi-Yau and Hom-finite.

Existence of n-cluster tilting objects in Higgs categories.

Applications to relative Ginzburg dg algebras and higher Auslander algebras.

Abstract

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) and Plamondon (2011) to arbitrary cluster algebras associated with quivers. A higher dimensional generalization is due to Guo (2011). Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, etc. The work of Geiss-Leclerc-Schr\"oer often yields Frobenius exact categories which allow us to categorify such cluster algebras. In this paper, we generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an $ n $-cluster tilting object in a Frobenius extriangulated category, namely the Higgs category (generalizing the Frobenius categories of Geiss-Leclerc-Schr\"oer), which is stably $ n $-Calabi--Yau and Hom-finite, arising from a left $ (n+1) $-Calabi--Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to $ n $-representation-finite algebras.