AGZT-Lectures on formal multiple zeta values

TL;DR

This paper explores the algebraic structure of formal multiple zeta values, linking Hoffman's basis conjecture to the free odd generation conjecture, and extends Brown's theorem to the formal setting.

Contribution

It demonstrates that Hoffman's basis conjecture follows from the free odd generation conjecture and develops a coaction framework for formal multiple zeta values.

Findings

Hoffman's basis conjecture is implied by the free odd generation conjecture.

A coaction on the algebra of formal multiple zeta values is constructed.

The proof of Brown's theorem is extended to the formal multiple zeta values context.

Abstract

Formal multiple zeta values allow to study multiple zeta values by algebraic methods in a way that the open question about their transcendence is circumvented. In this note we show that Hoffman's basis conjecture for formal multiple zeta values is implied by the free odd generation conjecture for the double shuffle Lie algebra. We use the concept of a post-Lie structure for a convenient approach to the multiplication on the double shuffle group. From this, we get a coaction on the algebra of formal multiple zeta values. This in turn allows us to follow the proof of Brown's celebrated and unconditional theorem for the same result in the context of motivic multiple zeta values. We need the free odd generation conjecture twice: at first it gives a formula for the graded dimensions and secondly it is a key to derive a lift of the Zagier formula to the formal context.