The depth structure of motivic multiple zeta values

TL;DR

This paper explores the algebraic structure of motivic multiple zeta values, constructing maps and exact sequences to understand their depth-graded properties and proposing conjectures for higher depths.

Contribution

It introduces new maps and exact sequences for depth-graded motivic multiple zeta values and connects these to conjectures about motivic Lie algebras and generation by totally odd parts.

Findings

Exact sequences established for depths two and three.

Reduction of the triple zeta values conjecture to a linear algebra isomorphism.

New proof that motivic double zeta values modulo zeta(2) are generated by totally odd parts.

Abstract

In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. And from these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and three. In higher depth we conjecture that there are exact sequences of the same type. And we will show from three conjectures about depth-graded motivic Lie algebra we can nearly deduce the exact sequences conjectures in higher depth. At last we give a new proof of the result that the modulo zeta(2)$ version motivic double zeta values is generated by the totally odd part. And we reduce the well-known conjecture that the modulo zeta (2) version motivic triple zeta values is generated by the totally odd part to an isomorphism conjecture in linear algebra.