Generalized twisted quantum doubles and the McKay Correspondence

We consider a class of quasiHopf algebras which we call generalized twisted quantum doubles. They are abelian extensions H = Cinverted right perpendicular (G) over bar inverted left perpendicular* infinity Cinverted right perpendicularGinverted left perpendicular (G is a finite group, (G) over bar a homomorphic image, and * denotes the dual algebra), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if G is a subgroup of SU2(C) then H exhibits an orbifold McKay Correspondence: certain fusion rules of H define a graph with connected components indexed by conjugacy classes of (G) over bar, each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of G stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when (G) over bar = 1. (C) 2010 Elsevier Inc. All rights reserved.