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Monadic cointegrals and applications to quasi-Hopf algebras

Abstract : For C a finite tensor category we consider four versions of the central monad, A1,…,A4 on C . Two of them are Hopf monads, and for C pivotal, so are the remaining two. In that case all Ai are isomorphic as Hopf monads. We define a monadic cointegral for Ai to be an Ai -module morphism 1→Ai(D) , where D is the distinguished invertible object of C .We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case C is braided, to an integral for the braided Hopf algebra L=∫XX∨⊗X in C studied by Lyubashenko (1995).Our main motivation stems from the application to finite dimensional quasi-Hopf algebras H . For the category of finite-dimensional H -modules, we relate the four monadic cointegrals (two of which require H to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called γ -symmetrised cointegrals in the pivotal case, for γ the modulus of H .For (not necessarily semisimple) modular tensor categories C , Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of C , in particular of SL(2,Z) on C(L,1) . In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of S and T which uses the monadic cointegral.
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Submitted on : Wednesday, April 22, 2020 - 2:46:35 AM
Last modification on : Friday, April 1, 2022 - 3:46:35 AM

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Johannes Berger, Azat M. Gainutdinov, Ingo Runkel. Monadic cointegrals and applications to quasi-Hopf algebras. J.Pure Appl.Algebra, 2021, 225, pp.106678. ⟨10.1016/j.jpaa.2021.106678⟩. ⟨hal-02550179⟩

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