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The multiple Zeta value algebra and the stable derivation algebra
, 2003
"... Abstract. The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the GrothendieckTeichmüller group. We shall show that there is a canonical surjective Qlinear map from the graded dual vector space of the ..."
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Abstract. The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the GrothendieckTeichmüller group. We shall show that there is a canonical surjective Qlinear map from the graded dual vector space of the stable derivation algebra over Q to the newzeta space, the quotient space of the subvector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upperbound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier’s talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the newzeta space with the ladic Galois image Lie algebra which is associated with so the Galois representation on the prol fundamental group of P1 − {0,1, ∞}.
Tannakian fundamental groups associated to Galois groups
 In Galois groups and fundamental groups
, 2003
"... Abstract. The goal of this paper is to give background and motivation for several conjectures of Deligne and Goncharov concerning the action of the absolute Galois group on the fundamental group of the thrice punctured line, and to sketch solutions, complete and partial, of several of them. A major ..."
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Abstract. The goal of this paper is to give background and motivation for several conjectures of Deligne and Goncharov concerning the action of the absolute Galois group on the fundamental group of the thrice punctured line, and to sketch solutions, complete and partial, of several of them. A major ingredient in these is the theory of weighted completion of pro nite groups. An exposition of weighted completion from the point of view of tannakian categories is included.
RELATIVE COMPLETIONS AND THE COHOMOLOGY OF LINEAR GROUPS OVER LOCAL RINGS
"... For a discrete group G there are two wellknown completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group U, defined over Q, together with a homomorphism ψ: G → U that is universal among maps from G into prounipotent Qgroups. To construct U, it suffices to conside ..."
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For a discrete group G there are two wellknown completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group U, defined over Q, together with a homomorphism ψ: G → U that is universal among maps from G into prounipotent Qgroups. To construct U, it suffices to consider the case where G is nilpotent; the general case is handled by taking the inverse limit of the Malcev completions of the G/Γ r G, where Γ • G denotes the lower central series of G. If G is abelian, then U = G ⊗ Q. We review this construction in Section 2. The second completion of G is the pcompletion. For a prime p, we set G∧p = lim G is the plower central series of G. If G/Γr pG, where Γ•p G is a finitely generated abelian group, then G∧p = Zp ⊗ G, where Zp is the ring of padic integers [3]. The group G∧p is a propgroup and each G/Γr pG is nilpotent provided H1(G, Fp) is finite dimensional. Both of these completions are instances of a general construction. Let k be a field. The unipotent kcompletion of a group G is a prounipotent kgroup Uk together with a homomorphism G → Uk. The group Uk is required to satisfy the obvious universal mapping property. The Malcev completion is the case k = Q and the pcompletion is the case k = Fp. This construction for other fields k is probably wellknown to the experts, but it does not seem to be in the literature. One reason to study such completions is that they may be used to gain cohomological information about the groups G and U. The restriction map H i (U, k) → Hi (G, k) is an isomorphism for i = 0, 1, and the map H2 cts (U, k) → H2 (G, k) is injective (the definition of H 2 cts will be recalled below). This allows one to obtain either a lower bound for dimk H2 (G, k) or an upper bound for dimk H2 cts(U, k). Unfortunately, the group Uk may be trivial (e.g., G perfect, or more generally, if H1(G, k) = 0). To circumvent this, Deligne suggested the notion of relative completion. Suppose that ρ: G → S is a representation of G in a reductive group S defined over k. Assume that the image
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"... Abstract. We get a canonical embedding from the spectrum of the algebra of multiple zeta values modulo π 2 into the graded version of the GrothendieckTeichmüller group by using relations of the Drinfel’d associator. On the other hand, it is known that the rational prol Galois image algebraic group ..."
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Abstract. We get a canonical embedding from the spectrum of the algebra of multiple zeta values modulo π 2 into the graded version of the GrothendieckTeichmüller group by using relations of the Drinfel’d associator. On the other hand, it is known that the rational prol Galois image algebraic group is embedded into the GrothendieckTeichmüller proalgebraic group for each prime l. Via these embeddings, we get two kinds of relationship between the spectrum of the algebra of multiple zeta values and the prol