An explicit fusion algebra isomorphism for twisted quantum doubles of finite groups

Abstract

We exhibit an isomorphism between the fusion algebra of the quantum double of an extraspecial p-group, where p is an odd prime, and the fusion algebra of a twisted quantum double of an elementary abelian group of the same order.  2004 Elsevier Inc. All rights reserved. Keywords: Fusion algebra; Twisted quantum double of finite group PreliminariesThis article extends [3] to include odd primes. By demonstrating that D(G) and D ω (E) have isomorphic fusion algebras, we provide another family of examples involving (untwisted) quantum doubles of nonabelian groups and twisted quantum doubles of abelian groups with nonabelian cocycles (in the sense of [8]). The existence of our isomorphism is a special case of a more general theorem in [9]. We give one such isomorphism explicitly in Theorem 4.2 for its potential use in applications (such as [2], e.g.).For the entirety of this work, p is an odd prime. Fix ε to be a primitive pth root of unity and fix η so that η p = ε. A delta with two indices will be the usual Kronecker delta. Also, δ x∈A = 1 if x ∈ A, 0 if not. All tensor products are over C unless otherwise noted.