纤维函子的酉变换
Unitary transformations of fibre functors
摘要：
本文研究了紧量子群的纤维函子Rep（G）-> Hilb之间的幺正自然变换（UPT）.对于纤维函子F_1，F_2，我们证明了UPT F_1 -> F_2及其修改的范畴同构于相应的双Hopf-Galois对象的有限维 *-表示范畴。本文利用Rep（A_G）范畴中的Frobenius代数，给出了可由UPT从正则纤维函子中得到的纤维函子以及UPT本身的构造性分类，其中A_G是紧量子群的对偶Hopf *-代数.作为例子，我们证明了量子图X的有限维量子同构是Rep（GX）上纤维函子之间的UPT，其中GX是X的量子自同构群.
英文摘要：
We study unitary pseudonatural transformations (UPTs) between fibre functors Rep(G) -> Hilb, where G is a compact quantum group. For fibre functors F_1, F_2 we show that the category of UPTs F_1 -> F_2 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(A_G), where A_G 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(G_X), where G_X is the quantum automorphism group of X.
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