The Lie coalgebra of multiple polylogarithms

TL;DR

This paper constructs a Lie coalgebra structure for multiple polylogarithms using Goncharov's coproduct, establishing new relations and connecting to existing algebraic frameworks.

Contribution

It introduces a novel Lie coalgebra based on Goncharov's coproduct, incorporating functional relations and linking to Goncharov's Bloch groups and models.

Findings

Defined a Lie coalgebra over any field

Established relations including inversion and shuffle relations

Connected the structure to Goncharov's Bloch groups and models

Abstract

We use Goncharov's coproduct of multiple polylogarithms to define a Lie coalgebra over an arbitrary field. It is generated by symbols subject to inductively defined relations, which we think of as functional relations for multiple polylogarithms. In particular, we have inversion relations and shuffle relations. We relate our definition to Goncharov's Bloch groups, and to the concrete model in weight less than 5 by Goncharov and Rudenko.