EXPLICIT ASSOCIATOR RELATIONS FOR MULTIPLE ZETA VALUES

Associators were introduced by Drinfel’d in [Dri91] as a monodromy representation of a Knizhnik-Zamolodchikov equation. Associators can be briefly described as formal series in two non-commutative variables satisfying three equations. These three equations yield a large number of algebraic relations between the coefficients of the series, a situation which is particularly interesting in the case of the original Drinfel’d associator, whose coefficients are multiple zetas values. In the first part of this paper, we work out these algebraic relations among multiple zeta values by direct use of the defining relations of associators. While well-known for the first two relations, the algebraic relations we obtain for the third (pentagonal) relation, which are algorithmically explicit although we do not have a closed formula, do not seem to have been previously written down. The second part of the paper shows that if one has an explicit basis for the bar-construction of the moduli space M0,5 of genus zero Riemann surfaces with 5 marked points at one’s disposal, then the task of writing down the algebraic relations corresponding to the pentagon relation becomes significantly easier and more economical compared to the direct calculation above. We discuss the explicit basis described by Brown and Gangl, which is dual to the basis of the enveloping algebra of the braids Lie algebra UB5. In order to write down the relation between multiple zeta values, we then remark that it is enough to write down the relations associated to elements that generate the bar construction as an algebra. This corresponds to looking at the bar construction modulo shuffle, which is dual to the Lie algebra of 5-strand braids. We write down, in the appendix, the associated algebraic relations between multiple zeta values in weights 2 and 3.