To any Hopf algebra H we associate two commutative Hopf algebras Hl1(H) and Hl2(H), which we call the lazy homology Hopf algebras of H . These Hopf algebras are built in such a way that they are “predual” to the lazy cohomology groups based on the so-called lazy cocycles. We establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy ho...

Tannaka reconstruction provides a close link between monoidal categories and (quasi-)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coquasibialgebra one can consider a natural monoidal category consisting of Hopf modules, and one can ...

We show that if a finite dimensional Hopf algebra over C has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group G and a factorization G = G+G− into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in the similar way. Our results can be used to explain ...

The main goal is to study the Hopf algebra structure on quivers. The main result obtained by C. Cibils and M. Rosso is improved. That is, in the case of infinite dimensional isotypic components it is shown that the path coalgebra kQ admits a graded Hopf algebra structure if and only if Q is a Hopf quiver. All nonisomorphic point path Hopf algebras and point co-path Hopf algebras are found. The ...

If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H → H⊗QSymk is defined, where QSymk denotes the Hopf algebra of quasisymmetric functions. This homomorphism generalizes the “internal comultiplication” on QSymk, and extends what Hazewinkel (in §18.24 of his “Witt vectors”) calls the Bernstein homomorphism. We construct...

In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H

Let k be a commutative ring, H a faithfully flat Hopf algebra with bijective antipode, A a k-flat right H-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants B = A , and we study the semisimplicity of the category of relative Hopf modules.

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in [9] and [10]. In this note, we first generalize a result of Majid [15] for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of [4].

The main result in this paper states that every strongly graded bialgebra whose component of grade 1 is a finite-dimensional Hopf algebra is itself a Hopf algebra. This fact is used to obtain a group cohomology classification of strongly graded Hopf algebras, with 1-component of finite dimension, from known results on strongly graded bialgebras.  2002 Elsevier Science (USA)

The aim of this section is to define some generalization of the notion of formal group. More precisely, we consider the analog of formal groups with coefficients belonging to a Hopf algebra. We also study some example of a formal group over a Hopf algebra, which generalizes the formal group of geometric cobordisms. Recently some important connections between the Landweber-Novikov algebra and th...