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

TL;DR

The paper introduces a new $ obreak \mathbb{Q}$-form for Racinet's cyclotomic double shuffle Lie algebra, motivated by congruence relations for level-$N$ multiple zeta values. It develops parallel algebraic frameworks for $N$-MPVs and $N$-CMZVs via Hopf algebras and constructs a congruent double shuffle Lie algebra $\frak{dmr}_0^{[N]}$ that is isomorphic to the cyclotomic form $\frak{dmr}_0^{\mu_N}$ over the cyclotomic field with a Hopf- and Lie-theoretic bridge given by an explicit isomorphism $\mathcal{F}$. A Galois descent mechanism is established: the invariant subspaces under $G_N$ recover the $ obreak \mathbb{Q}$-forms, yielding isomorphisms $ (\mathbb{Q}(\mu_N) \hat{\otimes}_\mathbb{Q} \frak{dmr}_0^{[N]})^{\widetilde{\Delta}_{G_N}} \simeq \frak{dmr}_0^{\mu_N}$ and its dual. These results illuminate how cyclotomic and congruent double shuffle structures interrelate and hint at arithmetic descent phenomena for motivic Lie algebras tied to level-$N$ MZVs.

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.

Theorems & Definitions (48)

• Theorem 1: Theorem \ref{['main theorem']}
• Theorem 2: Theorem \ref{['Galois Descent theorem']}
• Lemma 1.1: AK04Zha10
• Lemma 1.2
• proof
• Definition 1.3: Rac02
• Proposition 1.4: Rac02
• Lemma 2.1: K24
• Lemma 2.2
• proof