Generalized Twisted Quantum Doubles and the McKay Correspondence

TL;DR

This paper introduces generalized twisted quantum doubles, a class of quasi-Hopf algebras extending the quantum double construction, and demonstrates their connection to the orbifold McKay Correspondence for subgroups of SU(2).

Contribution

It generalizes the twisted quantum double construction and establishes a new link between these algebras and the orbifold McKay Correspondence for certain finite groups.

Findings

Fusion rules define graphs with ADE-type components.

Connection between fusion rules and McKay Correspondence.

Reduces to classical McKay when ar G = 1.

Abstract

We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions $H = \mb{C}[\bar{G}] \bowtie \mb{C}[G]$ ($G$ is a finite group and $\bar{G}$ 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 $G$ is a subgroup of $SU_2(\mb{C})$ then $H$ exhibits an orbifold McKay Correspondence: certain fusion rules of $H$ define a graph with connected components indexed by conjugacy classes of $\bar{G}$, 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 $\bar{G} = 1$.