Related papers: Frobenius-Schur theorem for $C^*$-categories
This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C…
We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical…
We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras…
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We…
We present a unified ring theoretic approach, based on properties of the Casimir element of a symmetric algebra, to a variety of known divisibility results for the degrees of irreducible representations of semisimple Hopf algebras in…
In this article, we will define two canonical cohomology theories for Hopf $C^*$-algebras and for Hopf von Neumann algebras (with coefficients in their bicomodules). We will then study the situations when these cohomologies vanish. The…
This paper is concerned with the structures introduced recently by the authors of the current paper concerning the multiplier Hopf $*$-graph algebras and also the Cuntz-Krieger algebras and their relations with the $C^*$-graph algebras, and…
The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding…
A classical theorem of I. Schur states that the degree of any irreducible complex representation of a finite group $G$ divides the order of $G/\mathscr{Z} G$, where $\mathscr{Z} G$ is the center $G$. This note discusses similar divisibility…
In the setting of von Neumann algebras, measurable quantum groupoids have successfully been axiomatized and studied by Enock, Vallin, and Lesieur, whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact quantum…
This paper studies Frobenius subalgebra posets in abelian monoidal categories and shows that, under general conditions--satisfied in all semisimple tensor categories over the complex field--they collapse to lattices through a rigidity…
By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup.
In this paper, we study the duality theory of Hopf $C^*$-algebras in a general ``Hilbert-space-free'' framework. Our particular interests are the ``full duality'' and the ``reduced duality''. In order to study the reduced duality, we define…
We show that given a rigid C*-tensor category, there is an equivalence of categories between normalized irreducible Q-systems, also known as connected unitary Frobenius algebra objects, and compact connected W*-algebra objects. Although…
The aim of this paper is to introduce and to investigate the analogues of torsors for compact quantum groups and to study their role in representation theory. Let A be a unitarizable Hopf *-algebra: we show that there is a category…
Categories of paths are a generalization of several kinds of oriented discrete data that have been used to construct $C^*$-algebras. The techniques introduced to study these constructions apply almost verbatim to the more general situation…
We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras.…
The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how So{\l}tan's quantum Bohr compactification can be…
We establish a formula for the L-theory spectrum of real $C^*$-algebras from which we deduce a presentation of the L-groups in terms of the topological K-groups, extending all previously known results of this kind. Along the way, we extend…
We present an axiomatic frame (in Prt I of this book) in which many results of the K-theory for C*-algebras are proved. Then we construct an example for this axiomatic theory (in Part II), which generalizes the classical theory for…