What question did this study set out to answer?

To explore quantum group representations at complex roots of unity and their structural implications.

February 25, 2026Open Access

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

Key Points

To explore quantum group representations at complex roots of unity and their structural implications.
Calculation of Tannakian center via Lusztig’s quantum Frobenius functor.
Analysis of the associated fiber category over the dual group.
Examination of properties when the group is simply connected and the root is of even order.
The fiber category is established as a finite, integral braided tensor category.
For simply connected groups with even roots, it qualifies as a modular tensor category.
A finite-dimensional quasitriangular quasi-Hopf algebra is identified, which relates to the tensor category in question.

Abstract

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

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