Unitary transformations of fibre functors
Abstract
We study unitary pseudonatural transformations (UPTs) between fibre functors Rep(G) -> Hilb, where G is a compact quantum group. For fibre functors F1, F2 we show that the category of UPTs F1 -> F2 and modifications is isomorphic to the category of finite-dimensional *-representations of the corresponding bi-Hopf-Galois object. We give a constructive classification of fibre functors accessible by a UPT from the canonical fibre functor, as well as UPTs themselves, in terms of Frobenius algebras in the category Rep(AG), where AG is the Hopf *-algebra dual to the compact quantum group. As an example, we show that finite-dimensional quantum isomorphisms from a quantum graph X are UPTs between fibre functors on Rep(GX), where GX is the quantum automorphism group of X.