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Double shuffle Lie algebra and special derivations

Benjamin Enriquez, Hidekazu Furusho

TL;DR

This work identifies and analyzes a deep link between Racinet’s double shuffle Lie algebra $\mathfrak{dmr}_0$ and a broader automorphism framework built from tangential automorphisms. By constructing a rich network of groups, semidirect products, and category-theoretic diagrams, the authors show that the stabilizer of a certain universal diagram equals $\mathsf{DMR}_0(\mathbf k)$ and derive a key inclusion and ultimately equality between stabilizers of two fundamental data objects $[\![\Delta^{\mathcal{W}}_{r,l}]\!]$ and $[\![\![\rho_{\mathrm{DT}}]\!]$ in the group $\mathcal{G}$. The approach uses an intricate blend of Lie-theoretic, Hopf-algebraic, and category-theoretic machinery, including PSTGA structures, explicit matrix realizations, and cohomological arguments to track invariants under involutions. The results clarify how senary relations and the Krivine–Kontsevich–Ihara-type structures interact with inertia-preserving automorphisms, and they provide a robust framework for comparing stabilizers across different diagrammatic presentations, contributing to a deeper understanding of the algebraic underpinnings of multiple zeta value relations and associated Lie algebras.

Abstract

Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ is a Lie subalgebra of the Lie algebra $\mathfrak{tder}$ of tangential derivations of the free Lie algebra with generators $x_0,x_1$, i.e. of derivations such that $x_1\mapsto 0$ and $x_0\mapsto [a,x_0]$ for some element $a$. We prove: (1) $\mathfrak{dmr}_0$ is contained in the Lie subalgebra $\mathfrak{sder}$ of $\mathfrak{tder}$ of special derivations, i.e. satisfying the additional condition that $x_\infty\mapsto [b,x_\infty]$ for some element $b$, where $x_\infty:=x_1-x_0$; (2) $\mathfrak{dmr}_0$ is stable under the involution of $\mathfrak{sder}$ induced by the exchange of $x_0$ and $x_\infty$. The first statement: (a) says that any element of $\mathfrak{dmr}_0$ satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion $\mathfrak{dmr}_0\subset \mathfrak{krv}_2$ (which was proved by Schneps in 2012 only conditionally to the truth of (1)).

Double shuffle Lie algebra and special derivations

TL;DR

This work identifies and analyzes a deep link between Racinet’s double shuffle Lie algebra and a broader automorphism framework built from tangential automorphisms. By constructing a rich network of groups, semidirect products, and category-theoretic diagrams, the authors show that the stabilizer of a certain universal diagram equals and derive a key inclusion and ultimately equality between stabilizers of two fundamental data objects and in the group . The approach uses an intricate blend of Lie-theoretic, Hopf-algebraic, and category-theoretic machinery, including PSTGA structures, explicit matrix realizations, and cohomological arguments to track invariants under involutions. The results clarify how senary relations and the Krivine–Kontsevich–Ihara-type structures interact with inertia-preserving automorphisms, and they provide a robust framework for comparing stabilizers across different diagrammatic presentations, contributing to a deeper understanding of the algebraic underpinnings of multiple zeta value relations and associated Lie algebras.

Abstract

Racinet's double shuffle Lie algebra is a Lie subalgebra of the Lie algebra of tangential derivations of the free Lie algebra with generators , i.e. of derivations such that and for some element . We prove: (1) is contained in the Lie subalgebra of of special derivations, i.e. satisfying the additional condition that for some element , where ; (2) is stable under the involution of induced by the exchange of and . The first statement: (a) says that any element of satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion (which was proved by Schneps in 2012 only conditionally to the truth of (1)).
Paper Structure (104 sections, 227 theorems, 712 equations)

This paper contains 104 sections, 227 theorems, 712 equations.

Table of Contents

  1. Groups and group morphisms
  2. The group inclusion $\mathcal{G}_{\mathrm{inert}} \subset (\mathcal{G},\circledast)$
  3. Tangential and inertia-preserving automorphisms
  4. Relation of $(\mathcal{G},\circledast)$ and $\mathcal{G}_{\mathrm{inert}}$ with outer tangential and inertia-preserving automorphisms
  5. The group $(\mathrm{exp}(\mathbb L^\wedge),\circledast)$, the subgroup $\mathsf{DS}(\mathbf k)$, and Lie algebras
  6. Lie algebra isomorphisms
  7. Summary of the results
  8. Relation with involutions
  9. Acknowledgments
  10. The group equality $\mathrm{Stab}_{\mathcal{G}}([\!\![\Delta^{\mathcal{W}}_{r,l}]\!\!])=\mathsf{DMR}_0(\mathbf k)$
  11. Statement of group equality $\mathrm{Stab}_{\mathcal{G}}([\!\![\Delta^{\mathcal{W}}_{r,l}]\!\!])=\mathsf{DMR}_0(\mathbf k)$ and plan of proof
  12. Statement of group equality $\mathrm{Stab}_{\mathcal{G}}([\!\![\Delta^{\mathcal{W}}_{r,l}]\!\!]) =\mathsf{DMR}_0(\mathbf k)$
  13. Categorical material
  14. Sketch of the proof of Thm. \ref{['thm:main']}
  15. Diagrams in $\mathcal{G}\operatorname{-}\mathbf{PSGA}$ and $\mathbf{PSGA}_{\mathcal{G}}$
  16. ...and 89 more sections

Key Result

Lemma 2

(a) $(\mathrm{exp}(\mathfrak{lie}_{\{0,1\}}^\wedge),\circledast)$ is a group, and the map $(\mathrm{exp}(\mathfrak{lie}_{\{0,1\}}^\wedge),\circledast)\to(\mathbf k^2,+)$, $g\mapsto ((g|e_1),(g|e_0))$ is a group morphism. (b) The map $(\mathbf k^2,+)\to(\mathrm{exp}(\mathfrak{lie}_{\{0,1\}}^\wedge),\

Theorems & Definitions (517)

  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 507 more