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Multiple polylogarithms, polygons, trees and algebraic cycles, preprint
, 2005
"... Abstract. We construct, for a field F and a natural number n, algebraic cycles in Bloch’s cubical cycle group of codimension n cycles in ( P 1 F \{1}) 2n−1 which correspond to weight n multiple polylogarithms with generic arguments if F ⊂ C. Moreover, we construct out of them a Hopf subalgebra in th ..."
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Cited by 14 (3 self)
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Abstract. We construct, for a field F and a natural number n, algebraic cycles in Bloch’s cubical cycle group of codimension n cycles in ( P 1 F \{1}) 2n−1 which correspond to weight n multiple polylogarithms with generic arguments if F ⊂ C. Moreover, we construct out of them a Hopf subalgebra in the BlochKriz cycle Hopf algebra χcycle. In the process, we are led to other Hopf algebras built from trees and polygons, which are mapped to χcycle. We relate the coproducts to the one for Goncharov’s motivic multiple polylogarithms and to the ConnesKreimer coproduct on plane trees and produce the associated Hodge realization for polygons. Contents
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 14 (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.
Algebraic cycles and motivic generic iterated integrals. arXiv:math.NT/0506370
"... Abstract. Following [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in [BK]. It will be shown that th ..."
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Cited by 4 (1 self)
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Abstract. Following [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.
Cycle complex over P1 minus 3 points : toward multiple zeta values cycles
, 2012
"... Abstract. In this paper, the author constructs a family of algebraic cycles in Bloch’s cycle complex over P1 minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p are in the cubical cycle group of codimension p in (P1 \{0, ..."
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Cited by 3 (3 self)
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Abstract. In this paper, the author constructs a family of algebraic cycles in Bloch’s cycle complex over P1 minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p are in the cubical cycle group of codimension p in (P1 \{0, 1,∞})× (P1 \ {1})2p−1 and are, in weight greater or equal to 2, naturaly extended as equidimensional cycles over over A1. This allows to consider their fibers at the point 1 and this is one of the main differences with the work of Gangl, Goncharov and Levin. Considering the fiber at 1 makes it possible to think of these cycles as corresponding to weight n multiple zeta values. After the introduction, the author recalls some properties of Bloch’s cycle complex and enlightens the difficulties on a few examples. Then a large section is devoted to the combinatorial situation involving the combinatoric of trivalent trees. In the last section, two families of cycles are constructed as solutions to a “differential system ” in Bloch’s cycle complex. One of this
1.2. Basic definitions and notation 4
, 2005
"... ABSTRACT. In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion. CONTENTS ..."
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ABSTRACT. In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion. CONTENTS
ALGEBRAIC CYCLES AND MOTIVIC ITERATED INTEGRALS II
, 806
"... Abstract. This is a sequel to [FJ]. It will give a more natural framework for constructing elements in the Hopf algebra χF of framed mixed Tate motives according to Bloch and Kˇriˇz [BK]. This framework allows us to extend the results of [FJ] to interpret all multiple zeta values (including the dive ..."
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Abstract. This is a sequel to [FJ]. It will give a more natural framework for constructing elements in the Hopf algebra χF of framed mixed Tate motives according to Bloch and Kˇriˇz [BK]. This framework allows us to extend the results of [FJ] to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of χF. It implies that the prounipotent completion of the torsor of paths on P 1 − {0, 1, ∞}, is a mixed Tate motive in the sense of [BK]. Also It allows us to interpret the multiple logarithm Li1,...,1(z1,..., zn) as an element of χF as long as the
LOW WEIGHT MULTIPLE ZETA VALUES CYCLES.
, 2013
"... Abstract. In a recent work, the author has constructed two families of algebraic cycles in Bloch cycle algebra over P 1 \ {0,1,∞} that are expected to correspond to multiple polylogarithms in one variable and have a good specialization at 1 related to multiple zeta values. This is a short presentati ..."
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Abstract. In a recent work, the author has constructed two families of algebraic cycles in Bloch cycle algebra over P 1 \ {0,1,∞} that are expected to correspond to multiple polylogarithms in one variable and have a good specialization at 1 related to multiple zeta values. This is a short presentation, by the way of toy examples in low weight ( � 5), of this contruction and could serve as an introduction to the general setting. Working in low weight also makes it possible to push (“by hand”) the construction further. In particular, we will not only detail the construction of the cycle but we will also associate to these cycles explicit elements in the bar construction over the cycle algebra and make as explicit as possible the “bottowleft ” coefficient of the Hodge realization periods matrix. That is, in a few relevant cases we will associated to each cycles an integral showing how the specialization at 1 is related to multiple zeta values. We will be particularly interested in a new weight 3 example corresponding to −2ζ(2,1).
ON MOTIVIC ITERATED INTEGRALS IN GENERIC
"... This is a survey on the joint paper [FJ] with A. Jafari. We will explain a combinatorial framework for motivic study of iterated integrals on the affine line which is an extension of [GGLI] and [GGL2]. We will show that under a certain genericity condition these combinatorial ob ..."
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This is a survey on the joint paper [FJ] with A. Jafari. We will explain a combinatorial framework for motivic study of iterated integrals on the affine line which is an extension of [GGLI] and [GGL2]. We will show that under a certain genericity condition these combinatorial ob
Hopf algebras in renormalization . . .
, 2005
"... In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion. ..."
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In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.