Braided Groups and Quantum Fourier Transform

Abstract

We show that acting on every finite-dimensional factorizable ribbon Hopf algebra H there are invertible operators S, T obeying the modular identities (ST)3 = λS2, where λ is a constant. The class includes the finite-dimensional quantum groups uq(g) associated to complex simple Lie algebras. We give the example of uq(sl(2)) at a root of unity in detail, as well as an example relating to anyons. The operator S plays the role of "quantum Fourier Transform" and acts naturally on H viewed by transmutation as a braided group H (a braided-cocommutative Hopf algebra in a braided category). It obeys S2 = s−1, where s is the antipode of H. The results follow as an application of previous category-theoretical constructions.