Log-Modular Quantum Groups at Even Roots of Unity and the Quantum Frobenius I

Abstract

We consider quantum group representations Rep(Gq) 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(Gq) is calculated via a twisting of Lusztig's quantum Frobenius functor Rep( Ǧ) → Rep(Gq), where Ǧ is a dual group to G. We then consider the associated fiber category Vect ⊗ Rep( Ǧ) Rep(Gq) over B Ǧ, 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 Vect ⊗ Rep( Ǧ) Rep(Gq), 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 [59].

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

“…To the best of our knowledge, the higher-rank analogues of the ADO invariants have not been worked out. 22 This gives us an opportunity to make new predictions for what they should be, (slN ) at odd root of unity. The restricted quantum group Uq(sl2) at even root of unity is defined in the same manner, but must be discussed with care.…”

“…The restricted quantum group Uq(sl2) at even root of unity is defined in the same manner, but must be discussed with care. See [22]. 23 The argument is the same as in the generalized volume conjecture [84], and the resulting invariants can often be formulated in terms of a "state integral" model a la [29] that involves states labeled by T C -valued variables [16,67,87].…”

“…The S and T matrices of a non-semisimple MTC and its semisimplification have different size, 3p − 1 and p − 1, respectively. See e.g [21,22]. for discussions on relations between two representation theories.…”

“…After this paper appeared, further evidence for (4.24) was presented in[86] 22. Part of the difficulty is that the detailed structure of projective modules was understood only recently using the tools of geometric representation theory[82,83], and mostly in the context of the small quantum…”

On the spectrum and support theory of a finite tensor category

“…The Tate side: the conditions (ii') and (iii') entering in Theorems B(c), C(b) and D(b) are strong conditions on T. The question of what finite tensor categories satisfy them is currently under intense study [6,18,39] and even the case of module categories of finite groups is still open. In these situations Theorems B(c), C(b) and D(b) give important information about cohomological support of elements of T computed with respect to Tate cohomology.…”