On bialgebras and Hopf algebras of oriented graphs

We define two coproducts for cycle-free oriented graphs, thus building up two commutative con- nected graded Hopf algebras, such that one is a comodule-coalgebra on the other, thus generalizing the result obtained previously for Hopf algebras of rooted trees.Comment: 7 pages, error on Proposition 1 corrected, one figure adde

Poisson bracket, deformed bracket and gauge group actions in Kontsevich deformation quantization

We express the difference between Poisson bracket and deformed bracket for Kontsevich deformation quantization on any Poisson manifold by means of second derivative of the formality quasi-isomorphism. The counterpart on star products of the action of formal diffeomorphisms on Poisson formal bivector fields is also investigated.Comment: 11 pages, one xypic figure. Minor changes on section I

Nested sums of symbols and renormalised multiple zeta functions

We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurin formula. Using a suitable holomorphic regularisation followed by a Birkhoff factorisation, we define renormalised nested sums of symbols which also satisfy stuffle relations. For appropriate symbols they give rise to renormalised multiple zeta functions which satisfy stuffle relations at all arguments. The Hurwitz multiple zeta functions fit into the framework as well. We show the rationality of multiple zeta values at nonpositive integer arguments, and a higher-dimensional analog is also investigated.Comment: Two major changes : improved treatment of the Hurwitz multiple zeta functions, and more conceptual (and shorter) approach of the multidimensional cas

Doubling bialgebras of rooted trees

The vector space spanned by rooted forests admits two graded bialgebra structures. The first is defined by A. Connes and D. Kreimer using admissible cuts, and the second is defined by D. Calaque, K. Ebrahimi-Fard and the second author using contraction of trees. In this article we define the doubling of these two spaces. We construct two bialgebra structures on these spaces which are in interaction, as well as two related associative products obtained by dualization. We also show that these two bialgebras verify a commutative diagram similar to the diagram verified D. Calaque, K. Ebrahimi-Fard and the second author in the case of rooted trees Hopf algebra, and by the second author in the case of cycle free oriented graphs

On matrix differential equations in the Hopf algebra of renormalization

We establish Sakakibara's differential equations in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.Comment: 22 pages, typos correcte

Confluence of singularities of differential equation: a Lie algebra contraction approach

We investigate here the confluence of singularities of Mathieu differential equation by means of the Lie algebra contraction of the Lie algebra of the motion group M(2) on the Heisenberg Lie algebra H(3). A similar approach for the Lam\'e equation in terms of the Lie algebra contraction of $SO_0(2,1)$ on the Lie algebra of the motion group M(2) is outlined

The combinatorics of Bogoliubov's recursion in renormalization

We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.Comment: improved version, 20 pages, CIRM 2006 workshop "Renormalization and Galois Theory", Org. F. Fauvet, J.-P. Rami