Geometric interpretation of double shuffle relation for multiple L-values

TL;DR

This paper provides a geometric interpretation of the double shuffle relations for multiple L-values, linking Enriquez' mixed pentagon equation to these relations and embedding related algebraic groups.

Contribution

It establishes that Enriquez' mixed pentagon equation implies the double shuffle relations for multiple L-values, extending previous results to cyclotomic cases.

Findings

Proves the implication of the mixed pentagon equation for double shuffle relations.

Constructs an embedding of the cyclotomic Grothendieck-Teichmuller group into the double shuffle group.

Extends previous results to cyclotomic analogues and general N cases.

Abstract

This paper gives a geometric interpretation of the generalized (including the regularization relation) double shuffle relation for multiple $L$-values. Precisely it is proved that Enriquez' mixed pentagon equation implies the relations. As a corollary, an embedding from his cyclotomic analogue of the Grothendieck-Teichmuller group into Racinet's cyclotomic double shuffle group is obtained. It cyclotomically extends the result of our previous paper and the project of Deligne and Terasoma which are the special case N=1 of our result.