In 1999 A. Connes and D. Kreimer have discovered a Hopf algebra structure on the Feynman graphs of scalar field theory. They have found that the renormalization can be interpreted as a solving of some Riemann — Hilbert problem. In this work the generalization of their scheme to the case of nonabelian gauge theories is proposed. The action of the gauge group on the Hopf algebra of diagrams is de...

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underly...

Recent work in perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we de...

We find the Hopf algebra U g,h dual to the Jordanian matrix quantum group GL g,h (2). As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: U ′ g,h (with three generators) and U (Z) (with one generator). The subalgebra U (Z) is a central Hopf subalgebra of U g,h. The subalgebra U ′ g,h is not a Hopf subalgebra and its coalgebra structure depends on both...

We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P . The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from differ...

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R,C) is a ribbon Hopf algebra, where R = (t ⊗ t)∆(t) and C = t. The element t is closely related to the topological ‘half-twist’, which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the categor...

We introduce and study a Hopf algebra containing the descent algebra as a sub-Hopf-algebra. It has the main algebraic properties of the descent algebra, and more: it is a sub-Hopf-algebra of the direct sum of the symmetric group algebras; it is closed under the corresponding inner product; it is cocommutative, so it is an enveloping algebra; it contains all Lie idempotents of the symmetric grou...

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underly...

In this paper, we construct explicitly a NCS system ([Z4]) Ω T ∈ (H GL) ×5 over the Grossman-Larson Hopf algebra H GL ([GL] and [F]) of rooted trees labeled by elements of a nonempty W ⊆ N of positive integers. By the universal property of the NCS system (NSym,Π) formed by the generating functions of certain NCSF’s ([GKLLRT]), we obtain a graded Hopf algebra homomorphism TW : NSym → H GL such t...