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

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 a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X = 1 + λa ≺ X and Y = 1 − λY a in A [[λ]] is provided, where (A, ≺, ) is a dendriform algebra. Then we present the solutions to these equations as an infinite product expansion of exponentials. Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.

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Abstract. We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees H CK by means of a natural H-bicomodule structure on H CK . This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.