Braided Groups and Quantum Fourier Transform Original Research Article

We show that acting on every finite-dimensional factorizable ribbon Hopf algebra H there are invertible operators image, image obeying the modular identities (image image)3 = λimage2, 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 image plays the role of "quantum Fourier Transform" and acts naturally on H viewed by transmutation as a braided group image (a braided-cocommutative Hopf algebra in a braided category). It obeys image2 = image−1, where image is the antipode of image. The results follow as an application of previous category-theoretical constructions.