264 research outputs found
On the bialgebra of functional graphs and differential algebras
We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators
Algebraic Birkhoff decomposition and its applications
Central in the Hopf algebra approach to the renormalization of perturbative
quantum field theory of Connes and Kreimer is their Algebraic Birkhoff
Decomposition. In this tutorial article, we introduce their decomposition and
prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give
some applications of this decomposition in the study of divergent integrals and
multiple zeta values.Comment: 39 pages. To appear in "Automorphic Forms and Langlands Program
On matrix differential equations in the Hopf algebra of renormalization
We establish Sakakibara's differential equations in a matrix setting for the
counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff
decomposition in any connected graded Hopf algebra, thus including Feynman
rules in perturbative renormalization as a key example.Comment: 22 pages, typos correcte
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
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