arXiv subjects
Publications and source records attributed to Mohamed Belhaj Mohamed.
We study the pre-Lie algebra of rooted trees $(\Cal T, \rightarrow)$ and we define a pre-Lie structure on its doubling space $(V, \leadsto)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted respectively by $(\Cal H', \star, Γ)$ and $(\Cal D', \bigstar, χ)$. We prove that $(\Cal D', \bigstar, χ)$ is a module-bialgebra on $(\Cal H', \star, Γ)$ and we find some relations between the two pre-Lie structures.
We study the pre-Lie algebra of specified Feynman graphs $\wt{V}_{\Cal T}$ and we define a pre-Lie structure on its doubling space $\wt{\Cal F}_{\Cal T}$. We prove that $\wt{\Cal F}_{\Cal T}$ is pre-Lie module on $\wt{V}_{\Cal T}$ and we find some relations between the two pre-Lie structures. Also, we study the enveloping algebras of two pre-Lie algebras denoted respectively by $(\wt {\Cal D'}_{\Cal T}, \bigstar, Φ)$ and $(\wt {\Cal H'}_{\Cal T}, \star, Ψ)$ and we prove that $(\wt {\Cal D'}_{\Cal T}, \bigstar, Φ)$ is a module-bialgebra on $(\wt {\Cal H'}_{\Cal T}, \star, Ψ)$.
In this article, we study the shuffle quadri-algebra H. We prove the existence of some relations between quadri-algebra laws which constitute shuffle product, the concatenation product and the deconcatenation coproduct. We also show that the underlying shuffle algebra has two module-algebra structures on itself.
In this article, we define a doubling procedure for the bialgebra of specified Feynman graphs introduced in a previous paper \cite {DMB}. This is the vector space generated by the pairs $(\bar Γ, \bar γ)$ where $\bar Γ$ is a locally $1PI$ specified graph of a perturbation theory $\Cal T$ with $\bar γ\subset \bar Γ$ locally $1PI$ and where $\bar Γ/ \bar γ$ is a specified graph of $\Cal T$. We also define a convolution product on the characters of this new bialgebra with values in an endomorphism algebra, equipped with a commutative product compatible with the composition. We then express in this framework the renormalization as formulated by A. Smirnov \cite [§8.5, 8.6] {Sm}, adapting the approach of A. Connes and D. Kreimer for two renormalization schemes: the minimal renormalization scheme and the Taylor expansion scheme. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure, by following the approach by P. Etingof \cite{PE} (see also \cite{RM}).
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.
We construct a Hopf algebra structure on the space of specified Feynman graphs of a quantum field theory. We introduce a convolution product and a semigroup of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.
We consider two interacting connected graded Hopf algebras, the former being a comodule-coalgebra on the latter. We show how to define analogues of Connes-Kreimer's renormalization group and Beta function, when the graduation operator is replaced by any biderivation coming from an infinitesimal character of the second Hopf algebra.