2025/05/04 by Benjamin Enriquez, Enriquez, Benjamin, Hidekazu Furusho +1 · 3 citations
Mathematics · Physics and Astronomy · #17B37 17B70 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2505.02265
openalex publication_date 2025/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Racinet's double shuffle Lie algebra \mathfrakdmr0 is a Lie subalgebra of the Lie algebra \mathfraktder of tangential derivations of the free Lie algebra with generators x0,x1, i.e. of derivations such that x1↦ 0 and x0↦ [a,x0] for some element a. We prove: (1) \mathfrakdmr0 is contained in the Lie subalgebra \mathfraksder of \mathfraktder of special derivations, i.e. satisfying the additional condition that x_∞↦ [b,x_∞] for some element b, where x_∞:=x1-x0; (2) \mathfrakdmr0 is stable under the involution of \mathfraksder induced by the exchange of x0 and x_∞. The first statement: (a) says that any element of \mathfrakdmr0 satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion \mathfrakdmr0⊂ \mathfrakkrv2 (which was proved by Schneps in 2012 only conditionally to the truth of (1)). We also derive the analogues of statements (a) and (b) respective to Racinet's ``double shuffle schemes'' DMRμ(\mathbf k) and to the Betti double shuffle group DMRB(\mathbf k) introduced in our earlier work.