galois extension for a compact quantum group
نویسنده
چکیده
The aim of this paper is to introduce the quantum analogues of torsors for a compact quantum group and to investigate their relations with representation theory. Let A be a Hopf algebra over a field k. A theorem of Ulbrich asserts that there is an equivalence of categories between neutral fibre functors on the category of finitedimensional A-comodules and A-Galois extensions of k. We give the compact quantum group version of this result. Let A be the Hopf ∗-algebra of representative functions on a compact quantum group. We show that there is an equivalence of categories between ∗-fibre functors on unitary A-comodules and A-∗-Galois extensions with a positive Haar measure. Such an A-∗-Galois extension has a C∗-norm, which furthermore can be taken as the upperbound of C∗-semi-norms. We then introduce the notion of Galois extension for a compact quantum group, for which measure theory can be deduced from topology. We construct universal Galois extensions, which enable us to find a nontrivial Galois extension for the unitary quantum group Uq(2).
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