论文标题
Monadic协整和用于准Hopf代数的应用
Monadic cointegrals and applications to quasi-Hopf algebras
论文作者
论文摘要
对于$ \ Mathcal {C} $一个有限张量类别，我们考虑了$ \ Mathcal {C} $上的中央单元，$ a_1，\ dots，a_4 $的四个版本。其中两个是Hopf Monads，对于$ \ Mathcal {C} $关键，其余两个也是如此。在这种情况下，所有$ a_i $都是hopf monads同构。我们为$ a_i $定义了一个单本的协调性，为$ a_i $ -module module $ \ mathbf {1} \ to a_i（d）$，其中$ d $是$ \ mathcal {c} $的杰出可逆对象。我们将Monadic协整与Shimizu（2019）介绍的分类合并性相关联，在$ \ Mathcal {C} $中，将编织构成编织，与编织的Hopf algebra $ \ Mathcal $ \ Mathcal {l} = \ Int^x x x x^x x^x x^x $ n $ n $ Lyubashenko（1995）。我们的主要动机源于对有限维数准代数$ h $的应用程序的应用。对于有限维$ h $模型的类别，我们将四个单一的合并曲（其中两个都需要$ h $保持关键）与四个现有的quasi-hopf代数的概念：通常的左/右/右concomintrals of Hausser and Nill（1994年），以及$ $ $ $ $ - 案例，对于$γ$，$ h $的模量。对于（不一定是半个）模块化张量类别$ \ MATHCAL {C} $，Lyubashenko在$ \ Mathcal {C} $的某些HOM空间上采取了表面映射类的动作$ \ MATHCAL {C}（\ MATHCAL {L}，\ MATHBF {1}）$。对于可分解的色带准Hopf代数，我们为使用Monadic Concomentegral的$ S $和$ t $的动作提供了简单的表达。
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 $\mathbf{1} \to A_i(D)$, 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 $γ$-symmetrised cointegrals in the pivotal case, for $γ$ 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 $\mathcal{C}(\mathcal{L},\mathbf{1})$. 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.
