Zinbiel algebras and multiple zeta values

TL;DR

This paper explores the algebraic structures underlying multiple zeta values using zinbiel algebras, constructing subalgebras and analyzing their relationships with motivic multiple zeta values through quotient maps.

Contribution

It introduces a new approach to study multiple zeta values via zinbiel algebra structures and constructs specific subalgebras with conjectured isomorphisms to motivic multiple zeta values.

Findings

Construction of commutative subalgebras within formal iterated integrals.

Identification of a quotient map relating these subalgebras to motivic multiple zeta values.

Conjecture that the map is generically an isomorphism.

Abstract

We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values.