An alternative $\mathbb{Q}$-form of the cyclotomic double shuffle Lie algebra

TL;DR

This paper introduces a new rational form of the cyclotomic double shuffle Lie algebra, showing how it relates to existing forms through Galois invariance, inspired by multiple zeta value relations.

Contribution

It provides an alternative $Q$-form of the cyclotomic double shuffle Lie algebra and establishes a Galois invariance theorem linking different $Q$-forms.

Findings

New $Q$-form of the algebra introduced

Galois invariance characterizes the relation between forms

Reconstruction of forms from each other under Galois action

Abstract

We present an alternative $\mathbb{Q}$-form for Racinet's cyclotomic double shuffle Lie algebra, inspired by the double shuffle relations among congruent multiple zeta values studied by Yuan and Zhao. Our main result establishes an invariance characterization theorem, demonstrating how these two $\mathbb{Q}$-forms can be reconstructed from each other under Galois action.