Rooted tree maps for multiple $L$-values from a perspective of harmonic algebras

TL;DR

This paper explores the structure of rooted tree maps related to multiple L-values, demonstrating their placement within the kernel of the evaluation map and analyzing their algebraic properties using harmonic algebra concepts.

Contribution

It introduces the diamond product as a modified harmonic product and establishes the antipode relationship for tau-conjugate rooted tree maps.

Findings

Rooted tree maps form a subspace of the kernel of the evaluation map.

The diamond product has specific algebraic properties.

Tau-conjugate rooted tree maps are antipodes.

Abstract

In this paper, we show the image of rooted tree maps themselves forms a subspace of the kernel of the evaluation map of multiple $L$-values. For its proof, we define the diamond product as a modified harmonic product and find its properties. We also show that $\tau$-conjugate rooted tree maps are their antipodes.