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

Abstract We consider quantum group representations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{Rep}(G_{q}) is calculated via a twisting of Lusztig’s quantum Frobenius functor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>̌</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">→</m:mo> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \operatorname{Rep}(\check{G})\to\operatorname{Rep}(G_{q}) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>̌</m:mo> </m:mover> </m:math> \check{G} is a dual group to 𝐺. We then consider the associated fiber category <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Vect</m:mi> <m:msub> <m:mo lspace="0.222em" rspace="0.222em">⊗</m:mo> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>̌</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>⁢</m:mo> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>̌</m:mo> </m:mover> </m:mrow> </m:math> B\check{G} , 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Vect</m:mi> <m:msub> <m:mo lspace="0.222em" rspace="0.222em">⊗</m:mo> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>̌</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mrow> <m:mi>Rep</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2026-0005_ineq_00