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Leftsymmetric algebras, or preLie algebras in geometry and physics
 Cent. Eur. J. Math
"... Abstract. In this survey article we discuss the origin, theory and applications of leftsymmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name preLie algebras is ..."
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Cited by 49 (11 self)
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Abstract. In this survey article we discuss the origin, theory and applications of leftsymmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name preLie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.
RotaBaxter algebras in renormalization of perturbative quantum field theory
 Fields Institute Communications v. 50, AMS
"... Abstract. Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularize ..."
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Cited by 35 (11 self)
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Abstract. Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. In this context the notion of Rota–Baxter algebras enters the scene. We review several aspects of Rota–Baxter algebras as they appear in other sectors also relevant to perturbative renormalization, for instance multiplezetavalues and matrix differential equations.
Factorization in quantum field theory: An exercise in Hopf algebras and local singularities
 Proceedings From Number Theory to Physics and Geometry, Les Houches March 2003, in press, arXiv:hepth/0306020
"... I discuss the role of Hochschild cohomology in Quantum Field Theory with particular ..."
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Cited by 26 (17 self)
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I discuss the role of Hochschild cohomology in Quantum Field Theory with particular
The Residues of Quantum Field Theory – Numbers we should know, contributed to the proceedings
 of the Workshop on Noncommutative Geometry and Number Theory, August 1822 2003, Max Planck Institut für Mathematik
"... ABSTRACT. We discuss in an introductory manner structural similarities between the polylogarithm and Green functions in quantum field theory. ..."
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Cited by 16 (7 self)
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ABSTRACT. We discuss in an introductory manner structural similarities between the polylogarithm and Green functions in quantum field theory.
An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
, 2005
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The structure of the Ladder InsertionElimination Lie algebra
 Commun. Math. Phys
"... Abstract. We continue our investigation into the insertionelimination Lie algebra LL of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson–Schwinger equations. We work out the relation to the classical infinite dim ..."
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Cited by 9 (5 self)
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Abstract. We continue our investigation into the insertionelimination Lie algebra LL of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson–Schwinger equations. We work out the relation to the classical infinite dimensional Lie algebra gl +(∞) and we determine the cohomology of LL. 1.
The uses of Connes and Kreimer’s algebraic formulation of renormalization theory
 Int. J. Mod. Phys. A
"... 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 and Zimmermann procedures for renormalizing Feynman amplitude ..."
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Cited by 6 (0 self)
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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 and Zimmermann procedures for renormalizing Feynman amplitudes. We discuss the outlook for a parallel simplification of computations in quantum field theory, stemming from the same algebraic approach.