Categorification of Skew-symmetrizable Cluster Algebras

Abstract

We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi–Yau category ${\mathcal{C}}$ endowed with the action of a finite group Γ, we construct a Γ-equivariant mutation on the set of maximal rigid Γ-invariant objects of ${\mathcal{C}}$ . Using an appropriate cluster character, we can then link these date to an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac–Moody groups, generalizing to the non simply-laced case several results of Geiß–Leclerc–Schröer.