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Coproducts and Cobrackets on Multiple Zeta Values

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Coproducts and Cobrackets on Multiple Zeta Values Coproducts and cobrackets on multiple zeta values Samuel Baumard Abstract. The aim of this talk is to describe a coproduct on a multiple zeta values introduced by A. Goncharov, and to discuss its connections with the Lie bracket on the double shue algebra. In a rst part, we dene the coproduct following Goncharov, and detail some of its properties in a particular space of formal zeta values. Then, the denition of the double shue Lie algebra is recalled, whose Lie bracket is in some way dual to a cobracket deduced from Goncharov's coproduct. The last part is devoted to introduce a recent version of a conjecture concerning Ihara's congruences in the double shue algebra. Contents 1. Goncharov's coproduct on multiple zeta values 1 2. Dual structures: the double shue Lie algebra 5 3. Ihara's congruences 7 References 11 1. Goncharov's coproduct on multiple zeta values 1.1. Denition of the coproduct. In the article [Gon05], Goncharov de- nes a coproduct on several spaces of iterated integrals, which gives in particular a coproduct on real multiple zeta values. Let us rst describe the iterated integrals he considers, and then the coproduct he builds on his version of formal multiple zeta values. Let (a0; : : : ; an+1) be a (n + 2)-uple of complex points, and let γ be a path connecting a0 to an+1. One can then dene the iterated integral Z def 1 dt1 dtn (1) Iγ (a0; a1; : : : ; an; an+1) = n ··· (2πi) ∆n,γ t1 − a1 tn − an where ∆n,γ is the set of all ordered n-uples (t1; : : : ; tn) of points on the path γ, provided that this integral converges (if it is not the case, it is however possible to give it a signication by regularizing it). If the ai's are in a number eld F , and having xed an embedding σ : F,! C, it is possible to map the integrals above into a graded algebra σ , which we shall not discuss here. The image P• (F ) of an integral depends no more on the choice of the path γ, and the elements thus dened obey several rules such as a shue product formula corresponding to the shue product of two iterated integrals. Note that if the ai's equal 0 or 1, one recovers the integral denition of classical multiple zeta values. By analogy with this algebra of iterated integrals, Goncharov denes for any set S a graded commutative algebra I•(S) as follows. a) The algebra has formal generators for any inte- I•(S) I(s0; s1; : : : ; sm; sm+1) ger and -uple of elements of . m > 0 (m + 2) si S b) For any pair of elements a; b 2 S, the generators I(a; ?; b) are equal and yield the unit of I•(S). 1 2 S. BAUMARD c) These elements satisfy the shue product formula, that is, if one xes two elements a and b of S and two integers m and n, then the identity X (2) I(a; s1; : : : ; sm; b) · I(a; sm+1; : : : ; sm+n; b) = I(a; sσ(1); : : : ; sσ(m+n); b) σ2Sm;n holds for any choice of elements si 2 S, where Sm;n denotes the set of all (m; n)- shues1. d) Every generator splits in the following way, called path I(s0; s1; : : : ; sm; sm+1) composition formula: for any x 2 S, one has m X . (3) I(s0; s1; : : : ; sm; sm+1) = I(s0; s1; : : : ; sk; x) · I(x; sk+1; : : : ; sm; sm+1) k=0 e) The element equals zero if (and otherwise). I(a; s1; : : : ; sm; a) m > 0 1 Each of the four latter conditions is the analogue of a similar property of iterated integrals. Moreover, the algebra has a natural grading L , I•(S) I•(S) = m 0 Im(S) obtained by giving to the generator the weight .> I(s0; s1; : : : ; sm; sm+1) m In the case of multiple zeta values, the set S is simply chosen to be f0; 1g, and it is then convenient to be able to express an element I(1; ::: ; 0) in terms of the I(0; ::: ; 1)'s. This is done by the following proposition, proved by Goncharov by induction on m: Proposition 1. If is an integer and are elements of , m > 0 a0; : : : ; am+1 S then m . (4) I(a0; a1; : : : ; am; am+1) = (−1) I(am+1; am; : : : ; a1; a0) Goncharov then denes a coproduct ∆ on the algebra I•(S) by giving the image of the generators (but the fact that this denition is coherent with the shue product formula is not obvious). A generator is sent by I(a0; a1; : : : ; am; am+1) ∆ on the element k X Y (5) I(ai0 ; ai1 ; : : : ; aik ; aik+1 ) ⊗ I(aip ; aip+1; : : : ; aip+1−1; aip+1 ) p=0 where the sum is over all (k + 2)-uples (i0; : : : ; ik+1) with k 2 [[0; m]] satisfying the conditions 0 = i0 < i1 < ··· ik < ik+1 = m + 1. 1.2. Interpretation in the Homann setting. The Homann notations are a useful way of encoding multiple zeta values. Let us recall them. First, the letter h denotes the algebra Qhx; yi of noncommutative polynomials in the two formal variables x and y with rational coecients. It contains the subalgebra 1 , which is generated (as an algebra) by the monomials h = Q ⊕ h y yi = xi−1y, and which in turn contains the subalgebra h0 = Q ⊕ x h y generated (as a vector space) by the monomials with . yi1 ··· yid i1 > 2 The multiple zeta values are encoded in the following manner. One denes a linear map ^ 0 which maps the monomial to the multiple zeta ζ : h ! R ys1 ··· ysd value X 1 (6) ζ(s1; : : : ; sd) = s1 sr n1 ··· nr n1>···>nr and which respects the units. The conditions dening h0 are precisely the ones needed to ensure the convergence of the above sum. Moreover, as mentioned before, 1 Recall that a (m; n)-shue is a permutation of the set [[1; m + n]] which reciprocal is in- creasing on the subsets [[1; m]] and [[m + 1; m + n]]. COPRODUCTS AND COBRACKETS ON MULTIPLE ZETA VALUES 3 there is a well-known representation of those multiple zeta values in terms of iterated integrals. Both representations of multiple zeta values give rise to an expression of the product of two multizetas as a sum of multizetas, which translates to an algebra structure on h0. More precisely, the shue product ø on h corresponds to the product of iterated integrals, and is given recursively by 0 0 0 (xiw) ø (xjw ) = xi (w ø (xjw )) + xj ((xiw) ø w ) (7) where xi 2 fx; yg. There is also an equivalent denition in terms of the sets Sm;n previously introduced. As for the stue product ∗ on h1, it corresponds to the product of series and is given by the recursive denition 0 0 0 0 (yiw) ∗ (yjw ) = yi (w ∗ (yjw )) + yj ((yiw) ∗ w ) + yi+j(w ∗ w ). (8) Both multiplication laws restrict to the subalgebra h0, and the linear map ζ^ becomes an algebra morphism for each of them. It is possible to extend the morphism ζ^ to a morphism from h1 (with the shue product) to R, and even from h to R, by means of a regularization formula (due to Furusho) involving shues and projection onto words that end with a y. One can also dene a formal version of these zeta values, considering the com- mutative algebra FZ generated by formal symbols Z(f) for any polynomial f in the x's and y's, subject to the following conditions: the symbols Z(f) must be li- near in f, and satisfy identities corresponding to the regularization and the shue and stue relations. For instance, one asks that for any divergent word w, the element Z(w) is equal to the element Z(w0) given after shue regularization of w, and that one has the identities Z(w) Z(w0) = Z(w ø w0) for any two words w and w0. Goncharov's coproduct should give a coproduct on the formal zeta values thus dened, or equivalently a coproduct on Qhx; yi compatible with the shue product and the relations of regularization and stue. If one follows directly Goncharov's denition (5), it yields to the following expression: let w = w1 ··· wn be a word in x and y; then k def X (9) ∆G(w) = wi1 ··· wik ⊗ Ø "i;p(wip+1 ··· wip+1−1) p=0 16i1<···<ik6n 06k6n where is a function that transforms the subword according "i;p wip+1 ··· wip+1−1 to the letters that surround it, coming from proposition 1 (by convention, i0 = 0 and ik+1 = n + 1). This denition is unfortunately not compatible with the stue relations, and the coproduct on Qhx; yi above dened does not give a coproduct on FZ. Although the study of the map dened in equation (9) is interesting in itself, there seems to remain some work to endow FZ with a Hopf algebra structure. 1.3. Coecients of pure tensors. Let h· ; ·i denote the natural scalar pro- duct on Qhx; yi for which the monomials form an orthonormal family, arising from the coecient maps on Qhx; yi; it also gives a scalar product on Qhx; yi⊗2 in the obvious way. It is natural to ask what the adjoint map of ∆G is or, in other words, what the product dual to this coproduct is. One nds a partial answer by computing coecients of words in the denition (9).
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