Brown-Zagier Relation for Associators

TL;DR

This paper proves that all associators satisfy a specific relation known as the Brown-Zagier relation, extending its validity to motivic multiple zeta values and employing cohomology and Li's method for the proof.

Contribution

It establishes the Brown-Zagier relation for all associators, broadening its applicability beyond previous proofs.

Findings

The relation holds for all associators.

The relation applies to motivic multiple zeta values.

Used cohomology theory and Li's method for proof.

Abstract

Francis Brown used a certain evaluation formula for multiple zeta values proved by Zagier to prove the injectivity of the homomorphism from the Motivic Galois group to the automorphism of fundamental group of projective line deleted three points. In this paper, we prove that any associators satisfies this relation. As a consequence, this relation holds for motivic multiple zeta values which is also proved in Brown's paper. We used certain cohomology theory and Li's method for the computation.