Motivic Double Shuffle

Soudères, Ismaël

doi:10.1142/s1793042110002995

Cited by 18 publications

(19 citation statements)

References 7 publications

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“…In this paper all the dual vector space means compact dual (linear functional whose support is finite dimension). Since motivic multiple zeta values satisfy double shuffle relations [16], [19], [21]. Any statement holds for the classical multiple zeta values which is deduced only by double shuffle relations also holds for the motivic multiple zeta values.…”

Section: Introductionmentioning

confidence: 93%

“…It's easy to show that gr D r A odd = gr D r A for r = 1, 2 from the results of [5], [11], [21]. The motivic Broadhurst-Kreimer conjecture (Conjecture 1.1) and its uneven part conjecture (Conjecture 1.2) suggest that gr D r A odd = gr D r A should also hold for r = 3.…”

Section: Introductionmentioning

confidence: 98%

“…In [5], Brown defined the motivic multiple zeta values by using an idea of Goncharov [12]. Motivic multiple zeta values satisfy the double shuffle relations [16], [19], [21], and there's a motivic Galois action on them. Their elements are Q-linear combinations of motivic multiple zeta values ζ m (n 1 , ..., n r ).…”

Section: Introductionmentioning

confidence: 99%

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The depth structure of motivic multiple zeta values

2018

Math. Ann.

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In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. From these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and three. In higher depth we conjecture that there are exact sequences of the same type. We will show from three conjectures about depth-graded motivic Lie algebra we can nearly deduce the exact sequences conjectures in higher depth. At last we give a new proof of the result that the modulo ζ m (2) version motivic double zeta values are generated by the totally odd part. We reduce the wellknown conjecture that the modulo ζ m (2) version motivic triple zeta values are generated by the totally odd part to an isomorphism conjecture in linear algebra.2010 M athematics Subject Classif ication. Primary 11F32, Secondary 11F67.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 98%

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The depth structure of motivic multiple zeta values

2018

Math. Ann.

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“…[18], but also [12], [16], [13], [8]), Goncharov (and co-authors) defined motivic multiple zeta values, which are (framed) mixed Tate motives with the motivic version ζ m (2) of ζ(2) being zero, and which form a Hopf algebra denoted MZ under the tensor product, with a coproduct defined explicitly by Goncharov ([13], foreshadowed by [12]). It is known that the motivic multiple zeta values satisfy relations (7.4)-(7.5) (see [13], for (7.5) see also [24]). Furthermore, F. Brown defined [4] a graded algebra comodule of motivic multiple zeta values H in which ζ m (2) is non-zero, and showed that H is non-canonically isomorphic to MZ ⊗ Q[ζ m (2)] and that Goncharov's period map (with values in a quotient of R only) can be lifted to a surjection H → → Z.…”

Section: 3mentioning

confidence: 99%

On the Broadhurst-Kreimer generating series for multiple zeta values

Carr¹,

Gangl²,

Schneps³

2015

Feynman Amplitudes, Periods and Motives

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Let F denote the free polynomial algebra F = Q s 3 , s 5 , s 7 ,. .. on non-commutative variables s i for odd i ≥ 3. The algebra F is weight-graded by letting sn be of weight n; we write Fn for the weight n part. In this paper we put a "special" decreasing depth filtration F = F 1 ⊃ F 2 ⊃ • • • ⊃ F d ⊃ F d+1 • • • on F , based on the period polynomials associated to cusp forms on SL 2 (Z). We define a lattice L of particular combinatorially defined subspaces of F , and conjecture that this lattice is distributive. Assuming this conjecture, we show that the dimensions of the weight n filtered quotients F d n /F d+1 n are given by the coefficients of the well-known Broadhurst-Kreimer generating series, defined by them to predict dimensions for the algebra of multiple zeta values. We end by explaining the expected relationship between F equipped with the special depth filtration and the algebras of formal and motivic multiple zeta values.

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“…This certainly occurs experimentally up to n = 9. The paper [19] by I. Soudères takes up this question in the context of motivic multiple zeta values.…”

Section: The Algebra Of Cell-zeta Valuesmentioning

confidence: 99%

The algebra of cell-zeta values

Brown¹,

Carr²,

Schneps³

2010

Compositio Math.

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In this paper, we introduce cell-forms on M 0,n , which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space M 0,n (R). We show that the cell-forms generate the top-dimensional cohomology group of M 0,n , so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cell-zeta values, which generate a Q-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.

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Motivic Double Shuffle

Cited by 18 publications

References 7 publications

The depth structure of motivic multiple zeta values

The depth structure of motivic multiple zeta values

On the Broadhurst-Kreimer generating series for multiple zeta values

The algebra of cell-zeta values

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