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

Abstract We consider quantum group representations Rep ⁡ (G q) Rep (Gₐ) 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) Rep (Gₐ) is calculated via a twisting of Lusztig’s quantum Frobenius functor Rep ⁡ (G ̌) → Rep ⁡ (G q) Rep (G) (Gₐ), where G ̌ G is a dual group to 𝐺. We then consider the associated fiber category Vect ⊗ Rep ⁡ (G ̌) Rep ⁡ (G q) Vectₑ₄₏ (₆) Rep (Gₐ) over B ⁢ G ̌ BG, 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 (also known as small quantum group) whose representations recover the tensor category Vect ⊗ Rep ⁡ (G ̌) Rep ⁡ (G q) <jats: inline-graphic xmlns: xlink="http: //www. w3. org/1999/xlink" xlink: href="graphic/jcrelle-2026-0005ᵢneq₀0