An explicit fusion algebra isomorphism for twisted quantum doubles of finite groups. (Q1763751)

Let \(p\) be an odd prime number. The author constructs an explicit isomorphism between the fusion algebras of the Drinfeld double \(D(G)\) of an extraspecial \(p\)-group \(G\) and the twisted Drinfeld double \(D^\omega(E)\), where \(E\) is an elementary Abelian group of order \(|E|=|G|\), and \(\omega\) is an appropriate 3-cocycle on \(E\). This extends previous work of the author [J. Algebra 259, No. 2, 494-511 (2003; Zbl 1023.17009)] for the case \(p=2\). Since \(G\) is a central extension of \(Z(G)\simeq\mathbb{Z}_p\) by an elementary Abelian group, the Drinfeld double \(D(G)\) is twist equivalent to the twisted Drinfeld double \(D^\omega(E)\), by the results in the reviewer's paper [J. Algebra 270, No. 1, 199--211 (2003; Zbl 1040.16027)]; this guarantees the existence of such fusion algebra isomorphisms.