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A primer of Hopf algebras
 Insitut des Hautes Études Scientifiques, IHES/M/06/04
, 2006
"... Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to H ..."
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Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with MilnorMoore’s theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions and the corresponding multiple zeta values. 1 Introduction.............................................
THE COMBINATORICS OF BOGOLIUBOV’S RECURSION IN renormalization
, 2008
"... We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the preLie Magnus expansion. Our results apply to any connected filtered Hopf a ..."
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We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the preLie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pronilpotency of the Lie algebra of infinitesimal characters.
Ambiguity theory, old and new
, 2008
"... This is a introductory survey of some recent developments of “Galois ideas” in Arithmetic, Complex Analysis, Transcendental Number Theory ..."
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This is a introductory survey of some recent developments of “Galois ideas” in Arithmetic, Complex Analysis, Transcendental Number Theory
MOTIVES: AN INTRODUCTORY SURVEY FOR PHYSICISTS
"... We survey certain accessible aspects of Grothendieck’s theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic th ..."
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We survey certain accessible aspects of Grothendieck’s theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic theory and theoretical physics.
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The
EJTP 10, No. 28 (2013) 1–8 Electronic Journal of Theoretical Physics What is a Quantum Equation of Motion?
"... Abstract: We apply combinatorial DysonSchwinger equations (in the context of the renormalization Hopf algebra) and noncommutative differential calculus to present a new interpretation from quantum motions. c © Electronic Journal of Theoretical Physics. All rights reserved. ..."
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Abstract: We apply combinatorial DysonSchwinger equations (in the context of the renormalization Hopf algebra) and noncommutative differential calculus to present a new interpretation from quantum motions. c © Electronic Journal of Theoretical Physics. All rights reserved.
Fixed point equations related to motion . . .
, 2009
"... In this paper we consider quantum motion integrals depended on the algebraic reconstruction of BPHZ method for perturbative renormalization in two different procedures. Then based on Bogoliubov character and BakerCampbellHausdorff (BCH) formula, we show that how motion integral condition on compo ..."
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In this paper we consider quantum motion integrals depended on the algebraic reconstruction of BPHZ method for perturbative renormalization in two different procedures. Then based on Bogoliubov character and BakerCampbellHausdorff (BCH) formula, we show that how motion integral condition on components of Birkhoff factorization of a Feynman rules character on ConnesKreimer Hopf algebra of rooted trees can determine a family of fixed point equations.