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30
Hopf algebras, from basics to applications to renormalisation, Rencontres Mathématiques de Glanon, and arXiv:math
"... These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a selfcontained introduction to the Hopfalgebraic techniques which appear in the work of A. Connes and D. Kreimer on renormalization in Quantum Field Theory [CK1], [CK2], ..."
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Cited by 35 (9 self)
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These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a selfcontained introduction to the Hopfalgebraic techniques which appear in the work of A. Connes and D. Kreimer on renormalization in Quantum Field Theory [CK1], [CK2], [BF]... Our point of view consists in revisiting
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 20 (8 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.
Birkhoff type decompositions and the Baker–Campbell– Hausdorff recursion
 Commun. Math. Phys
, 2006
"... Abstract. We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker–Cam ..."
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Cited by 16 (11 self)
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Abstract. We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker–Campbell–Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker–Campbell–Hausdorff recursion formula in the presence of a Rota–Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the evenodd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.
Renormalization of multiple zeta values
 J. Algebra
, 2006
"... Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special ..."
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Cited by 14 (10 self)
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Abstract. Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at nonpositive integers since the values are usually singular. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of IharaKanekoZagier on renormalization of MZVs with positive arguments. We further show that the important
A Lie theoretic approach to renormalization
 Comm. Math. Phys
"... Abstract. Motivated by recent work of Connes and Marcolli, based on the Connes– Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the ..."
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Cited by 12 (6 self)
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Abstract. Motivated by recent work of Connes and Marcolli, based on the Connes– Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on the fine properties of Hopf algebras and their associated descent algebras. Besides leading very directly to proofs of the main combinatorial properties of the renormalization procedures, the new techniques do not depend on the geometry underlying the particular case of dimensional regularization and the Riemann–Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.
On matrix differential equations in the Hopf algebra of renormalization
"... Abstract. We establish Sakakibara’s differential equations [Sa04] 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 exam ..."
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Cited by 7 (4 self)
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Abstract. We establish Sakakibara’s differential equations [Sa04] 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. Contents
Zimmermann type cancellation in the free Faà di Bruno algebra
 J. Funct. Anal
"... Krattenaler (BFK) in the context of noncommutative Lagrange inversion can be identified with the inverse of the incidence algebra of Ncolored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with noncommuting coe ..."
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Cited by 6 (0 self)
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Krattenaler (BFK) in the context of noncommutative Lagrange inversion can be identified with the inverse of the incidence algebra of Ncolored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with noncommuting coefficients and N noncommuting variables. As in the case of the Faà di Bruno Hopf algebra, there is an analogue of the Zimmermann cancellation formula. The summands of the (BFK) antipode can indexed by the depth first ordering of vertices on contracted planar trees, and the same applies to the interval partition antipode. Both can also be indexed by the breadth first ordering of vertices in the nonorder contractible planar trees in which precisely one nondegenerate vertex occurs on each level. 1.
Renormalization as a functor on bialgebras
 J. Pure Appl. Alg
"... Abstract. The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B) +), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When th ..."
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Cited by 5 (1 self)
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Abstract. The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B) +), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B) +), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S 1 (B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B) +) and the Faà di Bruno bialgebra of composition of series. The relation with the ConnesMoscovici Hopf algebra is given. Finally, the bialgebra S(S(B) +) is shown to give the same results as the standard renormalization procedure for the scalar field. 1.