Explicit associator relations for multiple zeta values

Abstract

Associators were introduced by Drinfel'd in as a monodromy representation of a KZ equation. Associators can be briefly described as formal series in two non-commutative variables satisfying three quations. 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 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 also discuss the explicit basis described by Brown and Gangl, which is dual to the basis of the enveloping algebra of the braids Lie algebra.