Action of finite groups on 2-Calabi-Yau categories and categorification of antisymetrizable cluster algebras

Laurent Demonet

We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A_2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type B_n [D]. We obtain also an interpretation of the MacKay correspondence in this setting : for a finite subgroup G of SL₂(C) containing the central element, G/Z(G) acts on the category of representations of the preprojective algebra of the Kronecker quiver. In this way, we obtain an equivalence of categories between the G/Z(G)-equivariant category and the category of representations of the preprojective algebra of the affine Dynkin diagram corresponding to G. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.

References

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