Dominique Manchon

Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series

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 by means of a natural H-bicomodule structure on the latter. 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.

A Magnus- and Fer-Type Formula in Dendriform Algebras

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.

A NONCOMMUTATIVE BOHNENBLUST-SPITZER IDENTITY FOR ROTA-BAXTER ALGEBRAS SOLVES BOGOLIUBOV'S RECURSION

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimers Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case, these identities can be understood as deriving from the theory of symmetric functions. Here, we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

Hopf Algebras in Renormalisation

Publisher Summary This chapter focuses on the Hopf algebras in renormalization. The chapter presents the Birkhoff decomposition. This chapter also explains the BCH approach to Birkhoff decomposition. This chapter discusses renormalized multiple zeta values, which is an application to number theory. The chapter explains the connected graded Hopf algebras. It concludes with a brief discussion on the renormalization group and the beta function.

A short survey on pre-Lie algebras

2 Preliminaries 4 2.1 Conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Fredholm index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 The Hodge-de Rham operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.1 The symbol and ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.2 Ellipticity and Fredholm properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 The index of the Hodge-de Rham operator . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 The definition of a spectral triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5.1 The Fredholm index in a spectral triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.2 Connes’ metric for spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Preliminaries 4 2.1 Conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Fredholm index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 The Hodge-de Rham operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.1 The symbol and ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.2 Ellipticity and Fredholm properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 The index of the Hodge-de Rham operator . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 The definition of a spectral triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5.1 The Fredholm index in a spectral triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.2 Connes’ metric for spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15We give an account of fundamental properties of pre-Lie algebras, and provide several examples borrowed from various domains of Mathematics and Physics : Algebra, Combinatorics, Quantum Field Theory and Numerical Analysis.

Shuffle Relations for Regularised Integrals of Symbols

AbstractWe prove shuffle relations which relate a product of regularised integrals of classical symbols

On matrix differential equations in the Hopf algebra of renormalization

\int^{reg} \sigma_i\, d\xi_i, i=1, \ldots, k

Poisson Bracket, Deformed Bracket and Gauge Group Actions in Kontsevich Deformation Quantization

to regularised nested iterated integrals: