Étude de l'algèbre de Lie double des arbres enracinés décorés

AbstractThe double Lie algebra LD of rooted trees decorated by a set D is introduced, generalising the construction of Connes and Kreimer. It is shown that it is a simple Lie algebra. Its derivations and its automorphisms are described, as well as some central extensions. Finally, the category of lowest weight modules is introduced and studied

Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.Comment: 16 page

On the structure and representations of the insertion-elimination Lie algebra

We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in \cite{CK}. It possesses a triangular structure \g = \n_+ \oplus \mathbb{C}.d \oplus \n_-, like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a "lowest weight" $\lambda \in \mathbb{C}$. We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible

Assessing the agro-environmental sustainability of organic mixed-crop dairy systems on the basis of a multivariate approach

Sustainable development calls upon the farming sector to commit itself to the transmission of natural resources to future generations. The INRA research team of Mirecourt studies the design of environmentally-friendly farming systems. The design of these systems is based on a multitude of objectives, and their evaluation is determined by a wide range of criteria. This work aims at determining the practical conditions for implementing agricultural systems considered to be sustainable from an environmental point of view. Two organic dairy systems considered to be environmentally friendly ex ante have been designed in partnership with the staff of the INRA research team of Mirecourt. A grazing dairy system and a mixed-crop dairy system are being experimentally tested at the system scale. The two systems have environmental and agricultural objectives. They are managed using multi-objective decision rules and are assessed on their biotechnical and practical properties, using a structured multiyear experimental design, completed by a model-based assessment. Assessment is oriented towards progressive and permanent re-designing of the systems in order to increase their environmental sustainability and feasibility at the practical level. Knowledge acquired from the two prototypes will then have to be validated on commercial farms

Combinatorial Hopf algebras in quantum field theory I

This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faa di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faa di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 rederives the related Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Section10 we describe the first. Then in Section11 we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Section 12 general incidence algebras are introduced, and the Faa di Bruno bialgebras are described as incidence bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained; this is the heart of the paper. The structure results for commutative Hopf algebras are found in Sections 14 and 15. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.Comment: 94 pages, LaTeX figures, precisions made, typos corrected, more references adde

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

Pea–wheat intercrops in low-input conditions combine high economic performances and low environmental impacts

Intensive agriculture ensures high yields but can cause serious environmental damages. The optimal use of soil and atmospheric sources of nitrogen in cereal–legume mixtures may allow farmers to maintain high production levels and good quality with low external N inputs, and could potentially decrease environmental impacts, particularly through a more efficient energy use. These potential advantages are presented in an overall assessment of cereal–legume systems, accounting for the agronomic, environmental, energetic, and economic performances. Based on a low-input experimental field network including 16 site-years, we found that yields of pea–wheat intercrops (about 4.5 Mg ha−1 whatever the amount of applied fertiliser) were higher than sole pea and close to conventionally managed wheat yields (5.4 Mg ha−1 on average), the intercrop requiring less than half of the nitrogen fertiliser per ton of grain compared to the sole wheat. The land equivalent ratio and a statistical analysis based on the Price\u27s equation showed that the crop mixture was more efficient than sole crops particularly under unfertilised situations. The estimated amount of energy consumed per ton of harvested grains was two to three times higher with conventionally managed wheat than with pea–wheat mixtures (fertilised or not). The intercrops allowed (i) maintaining wheat grain protein concentration and gross margin compared to wheat sole crop and (ii) increased the contribution of N2 fixation to total N accumulation of pea crop in the mixture compared to pea sole crop. They also led to a reduction of (i) pesticide use compared to sole crops and (ii) soil mineral nitrogen after harvest compared to pea sole crop. Our results demonstrate that pea–wheat intercropping is a promising way to produce cereal grains in an efficient, economically sustainable and environmentally friendly way

Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques

RésuméNous introduisons un foncteur de la catégorie des espaces tressés dans la catégorie des algèbres de Hopf tressées, associant à tout espace tressé V une algèbre de Hopf tressée d'arbres plans enracinés HP,R(V). Nous montrons que l'algèbre de Nichols de V est un sous-quotient de HP,R(V). Nous construisons un couplage de Hopf non dégénéré entre HP,R(V) et HP,R(V∗), généralisant ainsi l'un des résultats de [Bull. Sci. Math. 126 (2002) 193–239]. Lorsque le tressage de V est de la forme c(vi⊗vj)=qi,jvj⊗vi, nous obtenons une quantification des algèbres de Hopf d'arbres HDP,R introduites dans [Bull. Sci. Math. 126 (2002) 193–239 ; 126 (2002) 249–288]. Lorsque qi,j=qai,j, avec q une indéterminée et (ai,j)i,j la matrice de Cartan d'une algèbre de Lie semi-simple g, Uq(g+) est un sous-quotient de HP,R(V). Dans ce cas, nous effectuons le produit croisé de HP,R(V) avec un tore puis construisons le double de Drinfel'd D(HP,R(V)) de l'algèbre de Hopf ainsi obtenue. Nous montrons que Uq(g) est un sous-quotient de D(HP,R(V)).AbstractWe introduce a functor from the category of braided spaces into the category of braided Hopf algebras which associates to a braided space V a braided Hopf algebra of planar rooted trees HP,R(V). We show that the Nichols algebra of V is a subquotient of HP,R(V). We construct a Hopf pairing between HP,R(V) and HP,R(V∗), generalising one of the results of [Bull. Sci. Math. 126 (2002) 193–239]. When the braiding of c is given by c(vi⊗vj)=qi,jvj⊗vi, we obtain a quantification of the Hopf algebras HDP,R introduced in [Bull. Sci. Math. 126 (2002) 193–239; 126 (2002) 249–288]. When qi,j=qai,j, with q an indeterminate and (ai,j)i,j the Cartan matrix of a semi-simple Lie algebra g, then Uq(g+) is a subquotient of HP,R(V). In this case, we construct the crossed product of HP,R(V) with a torus and then the Drinfel'd quantum double D(HP,R(V)) of this Hopf algebra. We show that Uq(g) is a subquotient of D(HP,R(V))