  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.
