Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces

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 to Jacobi-finite quivers with potential (2009). Later, Plamondon generalized it to arbitrary cluster algebras associated with quivers (2009 and 2011). Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, … The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow to categorify such cluster algebras.

In previous work, we have constructed Higgs categories and relative cluster categories in the relative Jacobi-finite setting (arXiv:2109.03707). Higgs categories generalize the Frobenius categories used by Geiss-Leclerc-Schröer. In this talk, we give the construction of the Higgs category and of the relative cluster category in the relative Jacobi-infinite setting under suitable hypotheses. As in the relative Jacobi-finite case, the Higgs category is no longer exact but still extriangulated in the sense of Nakaoka-Palu (2019). We also give the construction of a cluster character in this setting.

This is a joint work with Chris Fraser and Bernhard Keller (arXiv:2307.12279).