We introduce a quantum double quasitriangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle φ. We also discuss D(Ug) as...

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...

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...

We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the RotaBaxter double construction, respectively Atkinson’s theorem. We also outline the extension...

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 ...

Since the Connes–Kreimer Hopf algebra was proposed, revisiting present quantum field theory has become meaningful and important from algebraic points. In this paper, the Hopf algebra in the cutting rules is constructed. Its coproduct contains all necessary ingredients for the cutting equation crucial to proving the perturbative unitarity of the S-matrix. Its antipode is compatible with the caus...

We give a rigorous proof that the (codimension one) Connes-Moscovici Hopf algebra HCM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the diffeomorphism group Diff(R). We construct a second bicrossproduct UCM equipped with a nondegenerate dual pairing with HCM. We give a natural quotient Hopf algebra kλ[Heis] of HCM and Hopf subalgebra Uλ(heis) of UCM which aga...