Quantum McKay Correspondence and Equivariant Sheaves on the Quantum Projective Line

Abstract

In this paper, using the quantum McKay correspondence, we construct the "derived category" of G-equivariant sheaves on the quantum projective line at a root of unity. More precisely, we use the representation theory of U_{q}sl(2) at root of unity to construct an analogue of the symmetric algebra and the structure sheaf. The analogue of the structure sheaf is, in fact, a complex, and moreover it is a dg-algebra. Our derived category arises via a triangulated category of G-equivariant dg-modules for this dg-algebra. We then relate this to representations of the quiver (\Gamma, \Om), where \Gamma is the A,D,E graph associated to G via the quantum McKay correspondence, and \Om is an orientation of \Gamma. As a corollary, our category categorifies the corresponding root lattice, and the indecomposable sheaves give the corresponding root system.