An extended version of a series lectures given at Bogota in december 2002. It consists presentation some aspects Connes' and Kreimer's work on renormalization the context general connected Hopf algebras, particular Birkhoff decomposition and, graded case, scattering-type formula.

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here theory functional identities noncommutative Rota–Baxter algebras which shown to encode, among others, this context Connes–Kreimer's Hopf algebra renormalization. Our results generalize seminal Cartier–Rota classical Spitzer-type commutative algebras. In classical, commutative,...

We provide a refined approach to the classical Magnus (Commun. Pure Appl. Math. 7:649–673, [1954]) and Fer expansion (Bull. Classe Sci. Acad. R. Belg. 44:818–829, [1958]), unveiling new structure by using language of dendriform pre-Lie algebras. The recursive formula for logarithm solutions equations X=1+λ ≺ X Y=1−λ Y ≻ in A[[λ]] is provided, where (A,≺,≻) algebra. Then we present these as an infinite product exponentials. Both formulae involve naturally associated with structure. Several...

Understanding the algebraic structure underlying a manifold with general affine connection is natural problem. In this context, A. V. Gavrilov introduced notion of framed Lie algebra, consisting bracket (the usual Jacobi vector fields) and magmatic product without any compatibility relations between them. work, we will show that an curvature torsion always gives rise to post-Lie algebra as well D -algebra. The notions together Gavrilov's special polynomials double exponential are revisited...

The existence of star products on any Poisson manifold M is a consequence Kontsevich's formality theorem, the proof which based an explicit formula giving quasi-isomorphism in flat case = R d .We propose here coherent choice orientations and signs order to carry case, i.e., prove that verifies indeed equation with all precised.

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide construction objects at play in regularity structures works by Bruned, Hairer and Zambotti (2019) Bruned Schratz (2022) for deriving general scheme dispersive PDEs low regularity. also explains how structure et al. cited above can be viewed as Butcher–Connes–Kreimer extraction-contraction Hopf algebras. start deforming various products via Taylor then we apply Guin–Oudom procedure...

We review the properties of transversality distributions with respect to submersions. This allows us construct a convolution product for large class on Lie groupoids. get unital involutive algebra \mathcal E_{r,s}'(G,\Omega^{1/2}) enlarging C^\infty_c(G,\Omega^{1/2}) associated any groupoid G . prove that -operators are operators by transversal distributions. also investigate microlocal aspects product. give sufficient conditions wave front sets compute and we show set two is essentially...

Dans l'article [6 Calaque , D. Ebrahimi-Fard K. Manchon ( 2011 ). Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution B-series . Advances in Appl. Math 47 2 ): 282 – 308 .[Crossref] [Google Scholar]], Damien Calaque, Kurusch et le premier auteur ont introduit un nouveau coproduit sur une algèbre commutative de forts d'arbres enracinés ℋ. L'algèbre Lie des éléments primitifs du dual gradué ℋ0 est munie d'une structure pré-Lie à gauche, notée ▷ qui...

We exhibit an internal coproduct on the Hopf algebra of finite topologies recently defined by second author, C. Malvenuto and F. Patras, dual to composition “quasi-ormoulds”, which are natural version J. Ecalle’s moulds in this setting. All these results displayed linear species formalism.

We define two coproducts for cycle-free oriented graphs, thus building up commutative connected graded Hopf algebras, such that one is a comodule-coalgebra on the other, generalizing result obtained in [2] algebras of rooted trees.

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally the combinatorics of set partitions, but also when studying compositions diffeomorphisms on vector spaces and manifolds, study cumulants moments probability theory. We construct commutative noncommutative explain how they give rise to Fa\`a di Bruno Hopf algebras. use language incidence algebras, along way provide a new description antipodes involving quasideterminants. discuss...