Hopf algebras and multiple zeta values in positive characteristic

TL;DR

This paper systematically studies the algebraic structures of multiple zeta values in positive characteristic, constructing Hopf algebras and resolving conjectures related to their algebraic properties.

Contribution

It introduces and develops the algebra and Hopf algebra structures of MZV's in positive characteristic, solving a problem posed by Deligne and Thakur and confirming Shi's conjectures.

Findings

Constructed the shuffle and stuffle algebra structures of MZV's in positive characteristic.

Established the Hopf algebra structures for these MZV's.

Solved a problem suggested by Deligne and Thakur and proved Shi's conjectures.

Abstract

Multiples zeta values (MZV's for short) in positive characteristic were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this paper we give a systematic study of algebraic structures of MZV's in positive characteristic. We construct both the stuffle algebra and the shuffle algebra of these MZV's and equip them with algebra and Hopf algebra structures. In particular, we completely solve a problem suggested by Deligne and Thakur \cite{Del17} in 2017 and establish Shi's conjectures \cite{Shi18}. The construction of the stuffle algebra is based on our recent work \cite{IKLNDP22}.