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Generalized Twisted Quantum Doubles and the McKay Correspondence

Abstract

We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions H = \mb{C}[\bar{G}] \bowtie \mb{C}[G] (GG is a finite group and Gˉ\bar{G} a homomorphic image), 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 GG is a subgroup of SU_2(\mb{C}) then HH exhibits an orbifold McKay Correspondence: certain fusion rules of HH define a graph with connected components indexed by conjugacy classes of Gˉ\bar{G}, each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of GG stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when Gˉ=1\bar{G} = 1.Comment: 5 figure

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