Monadic cointegrals and applications to quasi-Hopf algebras

Abstract

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1,\dots ,A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $1\to A_i(D)$, where $D$ is the distinguished invertible object of $\mathcal{C}$. We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L}={\int }^X{X}^{\vee }\otimes X$ in $\mathcal{C}$ 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 $\gamma$-symmetrised cointegrals in the pivotal case, for $\gamma$ the modulus of $H$. For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},1)$. 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.

arXiv comment59 pages; to appear in Journal of Pure and Applied Algebra, Volume 225, Issue 10, October 2021, 106678. Referees' comments taken into account

References in corpus (12)

• Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity
• Integral Theory for Quasi-Hopf Algebras
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• A quasi-Hopf algebra for the triplet vertex operator algebra
• Symplectic fermions and a quasi-Hopf algebra structure on ${\bar{U}}_{i}sl(2)$
• SL(2,Z)-action for ribbon quasi-Hopf algebras
• Modularization of small quantum groups
• Logarithmic Hennings invariants for restricted quantum sl(2)
• Renormalized Hennings Invariants and 2+1-TQFTs
• Modified traces for quasi-Hopf algebras
• Module traces and Hopf group-coalgebras
• Categorical aspects of cointegrals on quasi-Hopf algebras

Cited by in corpus (3)

• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• Modified traces for quasi-Hopf algebras
• Non-semisimple link and manifold invariants for symplectic fermions