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...
The goal of this paper is to give an elementary proof of the double shuffle relations directly for the Goncharov and Manin motivic multiple zeta values. The shuffle relation is straightforward, but for the stuffle, we use a modification of a method first introduced by Cartier for the purpose of proving stuffle for the real multiple zeta values. We will use both the representation of multiple zeta values on the moduli spaces of curve introduced by Goncharov and Manin and we will apply suitable blow-up sequences to the representation of multiple zeta values as integral over a cube.
Abstract. In a previous work, the author built two families of distinguished algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line minus three points.The goal of this paper is to show how these cycles induce well-defined elements in the H 0 of the bar construction of the cycle complex and thus generate comodules over this H 0 , that is a mixed Tate motives over the projective line minus three points.In addition, it is shown that out of the two families only one is needed at the bar construction level. As a consequence, the author obtains that one of the family gives a basis of the tannakian Lie coalgebra of mixed Tate motives over P 1 \ {0, 1, ∞} relatively to the tannakian Lie coalgebra of mixed Tate motives over Spec(Q). This in turns provides a new formula for Goncharov motivic coproduct, which really should be thought as a coaction. This new presentation is explicitly controlled by the structure coefficients of Ihara action by special derivation on the free Lie algebra on two generators.
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