Related papers: Weak Hopf algebras and quantum groupoids
We characterize braided commutative Yetter-Drinfeld $C^*$-algebras over weak Hopf $C^*$-algebras in categorical terms. Using this, we then study quotient type coideal subalgebras of a given weak Hopf $C^*$-algebra $\mathcal G$ and coideal…
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 investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with…
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an…
Let A be a finite dimensional Hopf algebra and (H, R) a quasitriangular bialgebra. Denote by H^*_R a certain deformation of the multiplication of H^* via R. We prove that H^*_R is a quantum commutative left H\otimes H^{op cop}-module…
An algebra with identities $[a,b]c=2a(bc)-2b(ac), a[b,c]=2(ab)c-2(ac)b$ is called weak Leibniz. We show that weak Leibniz operad is self-dual and is not Koszul. We establish that polarization of any weak Leibniz algebra is transposed…
Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing…
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful…
In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the…
Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is…
Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as…
We show that every partial representation of a connected Hopf algebra is global. Some interesting classes of partial representations of smash product Hopf algebras are studied, and a description of the partial "Hopf" algebra if the first…
The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by A. Van Daele and S. H. Wang. Using such…
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 establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak $C^*$-Hopf algebra introduced by G. Bohm, F. Nill and K. Szlachanyi (with an involutive…
The aim of this note is to show that weak relative hyperbolicity of a group relative to a subgroup (or relative hyperbolicity in the sense of Farb) does not imply any natural analogues of some well-known algebraic properties of ordinary…
The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of…
We put the known results on the antipode of a usual quasitriangular Hopf algebra into the framework of multiplier Hopf algebras. We illustrate with examples which can not be obtained using classical Hopf Algebras. The focus of the present…
Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper…
We give examples of Lie-Rinehart algebras whose enveloping algebra is not a full Hopf algebroid in the sense of Bohm and Szlachanyi. We construct these examples as quotients of a canonical Lie-Rinehart algebra over a Jacobi algebra which…