Modified traces and monadic cointegrals for quasi-Hopf algebras

Abstract

We introduce the notion of γ-symmetrized cointegrals for a finite-dimensional pivotal quasi-Hopf algebra H over a field k, where γ is the modulus of J In case H is unimodular and k is algebraically closed, we give explicit bijections relating them to non-degenerate left and right modified traces on the tensor ideal of projective H-modules in the (finite tensor) category of finite-dimensional left H-modules, generalizing previous Hopf-algebraic results from Beliakova-Blanchet-Gainutdinov.

Then we introduce monadic cointegrals in (pivotal) finite tensor categories. For a pivotal finite tensor category C, four versions (A₁, ..., A₄) of the so-called central Hopf monad exist. A monadic cointegral for A_i is a morphism of A_i-modules 1 -> A_i(D), where D is the distinguished invertible object of C; we relate them to Shimizu's categorical cointegral, and in the braided case to the integral of Lyubashenko's Hopf algebra ∫^(X in C) X* x X. If C is the category of modules over a pivotal Hopf algebra H, then one easily sees that the four monadic cointegrals are given by four notions of cointegrals for H, including γ-symmetrized cointegrals. We show that this relation, up to non-trivial isomorphisms, remains true if H is a quasi-Hopf algebra, i.e. we relate the cointegrals of Hausser and Nill and the γ-symmetrized cointegrals above to monadic cointegrals for the category of H-modules.

Finally, for a modular tensor category C, we concern ourselves with the projective SL(2,Z)-actions (on certain Hom-spaces in C) constructed by Lyubashenko. In the case that C is the category of modules over a factorizable ribbon quasi-Hopf algebra H, we derive a simple expression for the action of the S- and T-generators on the center of H using the monadic cointegral. Let now H be the quasi-Hopf algebra modification of the restricted quantum group of SL(2,Z) at a primitive 2p-th root of unity as constructed by Creutzig-Gainutdinov-Runkel, for an integer p ≥ 2. We show that Lyubashenko's action on the center of H agrees projectively with the SL(2,Z)-action on the center of the (original) restricted quantum group, as constructed by Feigin-Gainutdinov-Semikhatov-Tipunin.