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Noncommutative Hopf algebra of formal diffeomorphisms
, 2008
"... The subject of this paper are two Hopf algebras which are the noncommutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with ..."
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The subject of this paper are two Hopf algebras which are the noncommutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with noncommutative coefficients. Invertible series with noncommutative coefficients still form a group, and we interpret the corresponding new noncommutative Hopf algebra as an alternative to the natural Hopf algebra given by the coordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with noncommutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual noncommutative algebra there exists a natural coassociative coproduct defining a noncommutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a noncommutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semidirect coproduct of the previous Hopf algebras, and to series in
Rooted trees and an exponentiallike series
, 2002
"... This paper deals with a group of generalized power series associated to any augmented operad, focusing on the case of the PreLie operad. The solution of flow equations using the preLie structure on vector fields on an affine space gives rise to an interesting element of this group. 0 ..."
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Cited by 7 (4 self)
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This paper deals with a group of generalized power series associated to any augmented operad, focusing on the case of the PreLie operad. The solution of flow equations using the preLie structure on vector fields on an affine space gives rise to an interesting element of this group. 0
A stochastic calculus for multidimensional fractional Brownian motion with arbitrary Hurst index
, 2009
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An explicit rough path construction for continuous paths with arbitrary Hölder exponent
, 2009
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Combinatorial Hopf algebras
"... Abstract. We give a precise definition of “combinatorial Hopf algebras”, and we classify them in the four cases: associative or commutative, general or rightsided. For instance a cofreecocommutative combinatorial Hopf algebra is completely determined by its primitive part which is a preLie algebr ..."
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Cited by 4 (2 self)
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Abstract. We give a precise definition of “combinatorial Hopf algebras”, and we classify them in the four cases: associative or commutative, general or rightsided. For instance a cofreecocommutative combinatorial Hopf algebra is completely determined by its primitive part which is a preLie algebra. The classification gives rise to several good triples of operads. It involves the operads: dendriform, preLie, brace, GerstenhaberVoronov, and variations of them.
Groups of treeexpanded series
, 2008
"... In [3, 4] we introduced three Hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics. One of them, the algebra H α, is commutative, and is therefore the ring of coordinate functions of a proalgebraic group G α. The other two algebras, H e and H γ, are free non ..."
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Cited by 4 (1 self)
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In [3, 4] we introduced three Hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics. One of them, the algebra H α, is commutative, and is therefore the ring of coordinate functions of a proalgebraic group G α. The other two algebras, H e and H γ, are free noncommutative. Therefore their Abelian quotients are the coordinate rings of two proalgebraic groups G e and G γ. In this paper we describe explicitly these groups. Using two monoidal structures and a setoperad structure on planar binary trees, we show that these groups can be realized on formal series expanded over trees, and that the group laws are generalization of the multiplication and the composition of usual series in one variable. Therefore we obtain some new groups of invertible treeexpanded series and of treeexpanded formal diffeomorphisms respectively. The Hopf algebra describing the renormalization of the electric charge corresponds to the subgroup of treeexpanded formal diffeomorphisms formed of the translations, which fix the zero, by some particular treeexpanded series which remind the proper correlation functions in quantum field theory. In turn, the group of treeexpanded formal diffeomorphisms and some of its subgroups give rise to new Hopf algebras on trees. All the constructions are done in a general operadtheoretic setting, and then applied to the specific Duplicial operad on trees. All the
Multiplicative renormalization and Hopf algebras
, 2008
"... 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 Dyson’s formulas in quantum electrodynamics relating the ren ..."
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Cited by 4 (3 self)
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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 Dyson’s formulas in quantum electrodynamics relating the renormalized and bare proper functions via the renormalization constants and the analogous formulas for nonabelian gauge theories. In the latter case, we observe the crucial role played by Slavnov–Taylor identities. 1
Renormalization Hopf algebras and combinatorial groups
, 2009
"... These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory, held in Villa de Leyva (Colombia), July 2–20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many exa ..."
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Cited by 2 (0 self)
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These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory, held in Villa de Leyva (Colombia), July 2–20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the Lagrangian formalism to the EulerLagrange equation and the DysonSchwinger equation for Green’s functions. It poses the main problem of solving some nonlinear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar φ³ theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green’s functions given as perturbative series. The last