We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, ...

We introduce the concept of extended Hopf algebras and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structu...

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products are constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf a...

In this paper, we associate canonically a precyclic mod- ule to a regular multiplier Hopf algebra endowed with a group-like projection and a modular pair in involution satisfying certain con- dition

Consider the coradical filtration of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We show that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. We also sho...

Inspired by [3] we introduce the concept of extended Hopf algebra and consider their cyclic cohomology in the spirit of Connes-Moscovici [3, 4, 5]. Extended Hopf algebras are closely related, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscov...

We study forms of coalgebras and Hopf algebras (i.e. coalgebras and Hopf algebras which are isomorphic after a suitable extension of the base field). We classify all forms of grouplike coalgebras according to the structure of their simple subcoalgebras. For Hopf algebras, given a W ∗-Galois field extension K ⊆ L for W a finite-dimensional semisimple Hopf algebra and a K-Hopf algebra H, we show ...

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 by using classical Hopf algebras. The focus of the present paper lies on the class of the so-called G-cograded multiplier Hopf algebras. By doing so, we bring the results of quasitriangular Hopf group-coalgebras (a...

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. That is, the category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.