Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G)⊗K(G)) by the formula (∆(f))(p, q) = f(pq) for all f ∈ K(G) and p, q ∈ G. In this paper we consider multiplier Hopf algebras B (over C) such that there is an embeddi...

Among the most important topological constructions are Hopf fibrations. That they both play an important role in topology itself and have numerous physical applications lends credence to this statement. In this paper, I will discuss some of these applications as motivation for mathematicians to learn about the physical significance of the Hopf fibration in greater detail than that presented her...

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B)+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)+), the double symmetric algebra of B, into a comm...

We study the self-dual Hopf algebra HSP of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from HSP to the Hopf algebra of free quasi-symmetric functions FQSym given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to FQSym; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomor...

We notice that for a Hopf algebra H , its antipode S is both an epimorphism and a monomorphism from H to H in the category of Hopf algebras over a field. Together with the existence of Hopf algebras with non-injective or non-surjective antipode, this proves the existence of non-surjective epimorphisms and non-injective monomorphisms in the category of Hopf algebras. Using Schauenburg’s free Hop...

Given a nite graded poset with labeled Hasse diagram, we construct a quasi-symmetric generating function for chains whose labels have xed descents. This is a common generalization of a generating function for the ag f-vector de-ned by Ehrenborg and of a symmetric function associated to certain edge-labeled posets which arose in the theory of Schubert polynomials. We show this construction gives...

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes ...

The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra [3] can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras A = {A α } α∈π , ...

Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) = A⊗ A⊗C of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix fo...