Kenichi Shimizu

The pivotal cover and Frobenius–Schur indicators

Abstract In this paper, we introduce the notion of the pivotal cover C piv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius–Schur (FS) indicators in “non-pivotal” settings. For an object V ∈ C piv , the ( n , r ) -th FS indicator ν n , r ( V ) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators. Based on our framework, we also study the FS indicators of the “adjoint object” in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.

On indicators of Hopf algebras

Kashina, Montgomery and Ng introduced the n-th indicator νn(H) of a finite-dimensional Hopf algebra H and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of νn’s. In particular, we obtain the cyclotomic integrality of νn and a formula for νn of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra u(D, λ, µ) introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of uq(sl2) and show that if p and q are roots of unity of the same order, then up(sl2) and uq(sl2) are gauge equivalent if and only if q = p, where p and q are roots of unity of the same odd order.

The relative modular object and Frobenius extensions of finite Hopf algebras

Abstract For a certain kind of tensor functor F : C → D , we define the relative modular object χ F ∈ D as the “difference” between a left adjoint and a right adjoint of F . Our main result claims that, if C and D are finite tensor categories, then χ F can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension A / B of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of A / B . We also apply our results to obtain a “braided” version and a “bosonization” version of the result of Fischman et al.

Frobenius–Schur Indicator for Categories with Duality

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL2.

On the linear independency of monoidal natural transformations

Let

Frobenius-Schur indicators in Tambara-Yamagami categories

F, G: \mathcal{I} \to \mathcal{C}

The monoidal center and the character algebra

be strong monoidal functors from a skeletally small monoidal category

Some Computations of Frobenius–Schur Indicators of the Regular Representations of Hopf Algebras

\mathcal{I}

Non-degeneracy conditions for braided finite tensor categories

to a tensor category

On unimodular finite tensor categories

\mathcal{C}