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A Lie algebra associated with adjoint multiple zeta values

Takumi Anzawa

TL;DR

The paper investigates adjoint multiple zeta values (AdMZVs) and their adjoint double shuffle relations by constructing AdaDMR_0 and comparing it to Racinet's DMR_0 through tangent-space analysis, finding evidence against a direct isomorphism. It refines Jarossay's question by incorporating Hirose parity (V_str.prty) and defines a parity-constrained Lie structure inside the adjoint tangent space, using mould-theoretic tools to prove that a specific intersection forms a Lie subalgebra. The main result shows that addmr ∩ F2^{ad(x1)} ∩ V_str.prty yields a nontrivial Lie algebra, providing structural insight into AdMZV relations and a framework for a refined Kaneko–Zagier program via fiber products. The work combines advanced algebraic geometry (affine group schemes), Hopf-algebra dualities, and parity phenomena to offer new algebraic control over adjoint MZV relations and to chart directions for future lifting/conjecture formalisms. Computational data up to weight 11 support nuanced dimension matches and emphasize that a naïve isomorphism AdDMR_0 ≅ DMR_0 is unlikely, while the parity-constrained subspace remains rich for further structure.

Abstract

Jarossay (arXiv math.NT1412.5099) introduced adjoint multiple zeta values and, by using Racinet's dual formulation of the generating series of multiple zeta values, found $\mathbb{Q}$-algebraic relations among them, referred to as the \textit{adjoint double shuffle relations}. Additionally, Jarossay defined the affine scheme $\mathrm{AdDMR}_0$ determined by the adjoint double shuffle relations and posed a question whether $\mathrm{AdDMR}_0$ is isomorphic to Racinet's double shuffle group $\mathrm{DMR}_0$ (Publ. Math. Inst. Hautes Études Sci. (2002), no. 95). In this paper, we refine Jarossay's question by introducing the condition referred to as the adjoint conditions, and, based on this refinement, we study the corresponding Lie algebraic aspect. Within this framework, we construct the Lie algebra associated with the adjoint double shuffle relations by imposing Hirose's parity results.

A Lie algebra associated with adjoint multiple zeta values

TL;DR

The paper investigates adjoint multiple zeta values (AdMZVs) and their adjoint double shuffle relations by constructing AdaDMR_0 and comparing it to Racinet's DMR_0 through tangent-space analysis, finding evidence against a direct isomorphism. It refines Jarossay's question by incorporating Hirose parity (V_str.prty) and defines a parity-constrained Lie structure inside the adjoint tangent space, using mould-theoretic tools to prove that a specific intersection forms a Lie subalgebra. The main result shows that addmr ∩ F2^{ad(x1)} ∩ V_str.prty yields a nontrivial Lie algebra, providing structural insight into AdMZV relations and a framework for a refined Kaneko–Zagier program via fiber products. The work combines advanced algebraic geometry (affine group schemes), Hopf-algebra dualities, and parity phenomena to offer new algebraic control over adjoint MZV relations and to chart directions for future lifting/conjecture formalisms. Computational data up to weight 11 support nuanced dimension matches and emphasize that a naïve isomorphism AdDMR_0 ≅ DMR_0 is unlikely, while the parity-constrained subspace remains rich for further structure.

Abstract

Jarossay (arXiv math.NT1412.5099) introduced adjoint multiple zeta values and, by using Racinet's dual formulation of the generating series of multiple zeta values, found -algebraic relations among them, referred to as the \textit{adjoint double shuffle relations}. Additionally, Jarossay defined the affine scheme determined by the adjoint double shuffle relations and posed a question whether is isomorphic to Racinet's double shuffle group (Publ. Math. Inst. Hautes Études Sci. (2002), no. 95). In this paper, we refine Jarossay's question by introducing the condition referred to as the adjoint conditions, and, based on this refinement, we study the corresponding Lie algebraic aspect. Within this framework, we construct the Lie algebra associated with the adjoint double shuffle relations by imposing Hirose's parity results.

Paper Structure

This paper contains 9 sections, 36 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Algebraic setup
  3. Extended double shuffle relations and $\mathrm{DMR}_0$
  4. Adjoint double shuffle scheme $\mathrm{AdDMR}_0$
  5. The main theorem
  6. Proof of the main theorem
  7. Concluding remarks
  8. Proof of Lemma \ref{['lem:essential']}
  9. Computational data for $\mathfrak{dmr}$, $\mathfrak{addmr}$, $\mathfrak{addmr}\cap\mathfrak{F}_2^{\mathfrak{ad}(x_1)}$, and $\mathfrak{addmr} \cap \mathfrak{F}_2^{\mathfrak{ad}(x_1)} \cap V_{\mathrm{str.prty}}$

Key Result

Proposition 3.1

Let $T$ be a variable and $(R, Z_R)$ a pair of a $\mathbb{Q}$-algebra $R$ and an element $Z_R$ of $\mathop{\mathrm{Hom}}\nolimits_{\mathbb{Q}\text{-}\mathrm{\mathbf{Alg}}}(\mathfrak{h}^{0,\shuffle},R)$ with double shuffle conditions. Then the following is equivalent:

Theorems & Definitions (79)

  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1: Ihara-Kaneko-Zagier
  • Definition 3.2
  • Example 3.3
  • Conjecture 3.4
  • Theorem 3.5: Racinet
  • Definition 3.6: Racinet
  • Remark 3.7
  • Theorem 3.8: Racinet
  • ...and 69 more