Related papers: A Lie theoretic approach to renormalization
We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we…
We describe a new method of quantization of Lie bialgebras, based on a construction of Hopf algebras out of a cocommutative coalgebra and a braided comonoidal functor.
We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp…
We develop a new, intrinsic, computationally friendly approach to Lie coalgebras through graph coalgebras, which are new and likely to be of independent interest. Our graph coalgebraic approach has advantages both in finding relations…
This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting…
We study the notion of the Lie-holomorph of a Leibniz algebra, recently introduced by N. P. Souris as a generalisation of the classical holomorph construction for Lie algebras. We establish a connection between the Lie-holomorph…
We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.
We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages…
This paper is devoted to the study of renormalization of the quartic melonic tensor model in dimension (=rank) five. We review the perturbative renormalization and the computation of the one loop beta function, confirming the asymptotic…
We outline the proofs of several principal statements in conventional renormalization theory. This may be of some use in the light of new trends and new techniques (Hopf algebras, etc.) recently introduced in the field.
We review some important algebraic structures which appear in a priori remote areas of Mathematics, such as control theory, numerical methods for solving differential equations, and renormalization in Quantum Field Theory. Starting with…
We derive the existence of Hopf subalgebras generated by Green's functions in the Hopf algebra of Feynman graphs of a quantum field theory. This means that the coproduct closes on these Green's functions. It allows us for example to derive…
We construct explicit Drinfel'd twists for the generalized Cartan type $H$ Lie algebras in characteristic $0$ and obtain the corresponding quantizations and their integral forms. Via making modular reductions including modulo $p$ reduction…
A real Lie algebra defines by extension of scalars a complex Lie algebra that is isomorphic to its Galois conjugate. In this paper, we are interested in the converse property: given a complex Lie algebra that is isomorphic to its conjugate,…
We provide an algebraic framework to describe renormalization in regularity structures based on multi-indices for a large class of semi-linear stochastic PDEs. This framework is ``top-down", in the sense that we postulate the form of the…
We describe a formalism underlying the renormalization procedure and Batalin-Vilkoviski formalism. In the framework of this formalism, we give a mathematical definition of OPE-algebra and describe an additional natural structure which…
We study in this paper logarithmic derivatives associated to derivations on graded complete Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a…
This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate…
The Hopf algebra structure underlying Feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed. Recent progress is briefly summarized with an emphasis on further directions of research.
We generalize several important results from the perturbation theory of linear operators to the setting of semisimple orthogonal symmetric Lie algebras. These Lie algebras provide a unifying framework for various notions of matrix…