The Multiple Zeta Value Algebra and the Stable Derivation Algebra

Abstract

The MZV algebra is the graded algebra over $\mathbf{Q}$ generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the Grothendieck–Teichmüller group. We shall show that there is a canonical surjective $\mathbf{Q}$-linear map from the graded dual vector space of the stable derivation algebra over $\mathbf{Q}$ to the new-zeta space, the quotient space of the sub-vector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upper-bound 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 new-zeta space with the l-adic Galois image Lie algebra which is associated with the Galois representation on the pro-l fundamental group of $\mathbf{P}\frac{1}{\mathbf{Q}}-\{0,1,\infty \}$.