We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group G by an abelian group, with 3-cocycle inflated from a 3-cocycle on G. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two main applications: first, if G is an extra-special 2-group of width at least 2, we show that the quantum double of G twisted by a 3-cocycle ω is gauge equivalent to a twisted quantum double of an elementary abelian 2-group if, and only if, ω 2 is trivial; second, we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8, and classify the braided tensor equivalence classes of these quasi-triangular quasi-bialgebras. It turns out that there are exactly 20 such equivalence classes.

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We consider a class of quasiHopf algebras which we call generalized twisted quantum doubles. They are abelian extensions H = C[Ḡ] * C[G] (G is a finite group,Ḡ 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 SU 2 (C) then H exhibits an orbifold McKay Correspondence: certain fusion rules of H define a graph with connected components indexed by conjugacy classes ofḠ, 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 whenḠ = 1.

Let D ω (G) be the twisted quantum double of a finite group, G, where ω ∈ Z 3 (G, C *). For each n ∈ N, there exists an ω such that D(G) and D ω (E) have isomorphic fusion algebras, where G is an extraspecial 2-group with 2 2n+1 elements, and E is an elementary abelian group with |E| = |G|. Res G C K α=χ⊗ψ M (1, α) ⊕ M (−1, α) 6. M (K, χ) ⊗ M (L, ψ) = Res Q ξ=Res Q χ⊗Res Q ψ M (KL, ξ) 7. M (K, χ) ⊗ M (, Λ) = M (K, λ 1) ⊕ M (K, λ −1) 8. M (K, χ) ⊗ M (K, λ) = M (1, Λ) ⊕ M (−1, Λ) 9. M (K, χ) ⊗ M (L, λ) = M (KL, λ 1) ⊕ M (KL, λ −1) 10. M (1 , Λ) ⊗ M (2 , Λ) = α∈ G M (1 2 , α) 11. M (1 , Λ) ⊗ M (K, λ 2) = χ∈ C K M (K, χ) 12. M (K, λ 1) ⊗ M (K, λ 2) = α(g K)= 1 2 M (1, α) β(g K)=− 1 2 M (−1, β) 13. M (K, λ 1) ⊗ M (L, λ 2) = χ∈ C KL M (KL, χ) Proof: We will prove only a few cases to give the reader an idea of the straightforward proof. Full details require properties outlined in Section 5.1 and can also be found in [7]. Case 1: Consider the trace of the element (x ⊗ e g) on the left hand side of the equation. Via the coproduct, ∆, this element acts with trace zero unless g = 1 2. In this case, the trace becomes α(x)β(x) = (α ⊗ β)(x). Cases 2 and 10 are similar. Case 4: Consider the trace of the element (g K ⊗ e g K) on the left hand side. The only term in ∆ (g K ⊗ e g K) that has nonzero action is (g K ⊗ e 1) ⊗ (g K ⊗ e 1 g K). Thus the trace is α(g K)λ 2 (g K) if 1 = 1 and α(g K)λ − 2 (g K) if 1 is −1. Case 5: Consider the trace of (x ⊗ e 1 G) on the left hand side. There are two terms that have nonzero action. If g K has order 2, then the terms are (x ⊗ e g K) ⊗ (x ⊗ e g K) and (x ⊗ e g K z) ⊗ (x ⊗ e g K z). Hence (x ⊗ e 1 G) acts as 2χ(x)ψ(x)δ x∈C K. This character thus decomposes into two representations, each of

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