Quantum Frobenius and modularity for quantum groups at arbitrary roots of 1

Abstract

We consider quantum group representations Rep(G_q) for a semisimple algebraic group G at a complex root of unity q. Here we allow q to be of any order. We first show that the Tannakian center in Rep(G_q) is calculated via a twisting of Lusztig's quantum Frobenius functor Rep(H) -> Rep(G_q), where H is a dual group to G. We then consider the associated fiber category Rep(G_q)_{small} = Vect\otimes_{Rep(H)} Rep(G_q) over BH, and show that this fiber is a finite, integral braided tensor category. Furthermore, when G is simply-connected and q is of even order, the fiber in question is shown to be a modular tensor category. Finally, we exhibit a finite-dimensional quasitriangular quasi-Hopf algebra (aka, small quantum group) whose representations recover the tensor category Rep(G_q)_{small}, and we describe the representation theory of this algebra in detail. At particular pairings of G and q, our quasi-Hopf algebra is identified with Lusztig's original finite-dimensional Hopf algebra from the 90's. This work completes the author's project from arXiv:1812.02277.