Multiple Logarithms, Algebraic Cycles and Trees

Gangl, Herbert; Goncharov, A. B.; Levin, A.

doi:10.1007/978-3-540-30308-4_16

Cited by 8 publications

(18 citation statements)

References 6 publications

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“…Still, thanks to the universal property mentioned at the end of subsection 1.5, Hopf algebras of rooted trees serve as an excellent role model for various questions and, moreover, yield most interesting links to different branches of mathematics [13,21]. In the present paper, we will be mainly concerned with Hopf algebras of trees.…”

Section: Tree-like Structures and Variations On A Themementioning

confidence: 99%

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Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology

Bergbauer

Kreimer

2006

Physics and Number Theory

224

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Section: Tree-like Structures and Variations On A Themementioning

confidence: 99%

“…The solution of (21) can be described by c 0 = I and Proof. This is an easy induction using the following argument: Let τ be a given tree in c n and let its root o be decorated by something in degree m. According to (22)…”

Section: Combinatorial Dyson-schwinger Equationsmentioning

confidence: 99%

Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology

Bergbauer

Kreimer

2006

Physics and Number Theory

224

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“…In this section, we sketch how to associate an integral to the cycle L 0 011 . We directly follow the algorithm described in [GGL09][Section 9] and put in detailed practice in [GGL07]. There will be no general review of the direct Hodge realization from Bloch-Kriz motives [BK94][Section 8 and 9].…”

Section: 4mentioning

confidence: 99%

“…Relating the Bloch-Kriz approach to the explicit algorithms described in [GGL09,GGL07] and the application to our particular family of cycles L 0 W will be the topic of a future paper, as it requires, in particular, a family L 0 W B of elements in H 0 (B (N • X )) not yet at our disposal.…”

Section: 4mentioning

confidence: 99%

“…This leads, by specialization at x 0 = 1, to an algebraic cycle related to the (single) zeta value ζ(n). Algebraic cycles corresponding to multiple logarithm away from the point 1 have been constructed in [GGL07,GGL09] but these cycles cannot be specialized at the point 1. The algebraic cycles L 0 W given by Theorem 5.8 satisfy the following properties which are also satisfied by the algebraic cycles given by Theorem 6.4 (with the appropriate changes in notations).…”

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confidence: 99%

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The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

Soudères

2016

Journal of Pure and Applied Algebra

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Abstract. In this paper, we construct a family of algebraic cycles in Bloch's cycle complex over P 1 minus three points, which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p belong to the cubical cycle group of codimension p in (P 1 \ {0, 1, ∞}) × (P 1 \ {1}) 2p−1 and in weight greater than or equal to 2, they naturally extend as equidimensional cycles over A 1 .Thus, we can consider their fibers at the point 1. This is one of the main differences with the work of Gangl, Goncharov and Levin. Considering the fiber of our cycles at 1 makes it possible to view these cycles as those corresponding to weight n multiple zeta values which are viewed here as the values at 1 of multiple polylogarithms.After the introduction, we recall some properties of Bloch's cycle complex, and explain the difficulties on a few examples. Then a large section is devoted to the combinatorial situation, essentially involving the combinatorics of trivalent trees in relation with the structure of the free Lie algebra on two generators. In the last section, two families of cycles are constructed as solutions to a "differential system" in Bloch's cycle complex. One of these families contains only cycles with empty fiber at 0; these correspond to multiple polylogarithms. Lie coalgebra generating its 1-minimal model is isomorphic to the above Tannakian Lie coalgebra. ContentsIn order to obtain a motive, that is a comodule, it is enough to have an element in the relevant Tannakian Lie coalgebra. Then the motive is the comodule cogenerated by this element. Since the Tannakian Lie coalgebra is a "H 0 modulo product", we write one of its elements as a class [L B ] in this H 0 . This class can be represented in the bar construction over N • by an element L B . The element L B is essentially determined by its projection onto its tensor degree 1 part, which gives an algebraic cycle L in N • . The cycle L has a decomposable boundary because the element L B leads to a class in the H 0 .Hence a first step toward explicit motives via algebraic cycles is to build algebraic cycles in N • having a decomposable boundary. The main result of this paper (Theorem 5.8) provides such algebraic cycles denoted by L 0 W ; here the indexing set W consists in Lyndon words. It is shown in [Sou14] that the motives attached to these cycles generate the Lie coalgebra associated to the Deligne-Goncharov fundamental group.However lifting algebraic cycles to objects in the bar construction requires, in general, a strong combinatorial or algebraic control of the boundaries involved. In our context, this control is insured (Theorem 5.8) by the algebraic and combinatorial structure of Ihara's special derivations studied at Proposition 4.27.The next steps in order to obtain motivic multiple zeta values are:(1) Our main theorem makes it possible to build a commutative differential graded algebra morphism between the cobar construction over the Lie coalgebra associated to Ihara's special derivations (combinatorial structure...

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Factorization in Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities

Kreimer

Frontiers in Number Theory, Physics, and Geometry II

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Multiple Logarithms, Algebraic Cycles and Trees

Abstract: This is a short exposition-mostly by way of the toy models "double logarithm" and "triple logarithm"-which should serve as an introduction to the article [3] in which we establish a connection between multiple polylogarithms, rooted trees and algebraic cycles.

Cited by 8 publications

References 6 publications

Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology

Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology

The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

Factorization in Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities

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