A stabilizer interpretation de double shuffle Lie algebras

We recall the main results of double shuffle theory: the cyclotomicanalogues of MZVs (of order $N\geq 1$) satisfy a collection of relations arising from the study of their combinatorics, and also from their identifications with periods. The scheme arising from these relations is a torsor Under a prounipotent algebraic group $\mathrm{DMR}_0$.  This is a subgroup of the group $\mathrm{Out}^*$ of invariant tangential outer automorphisms of a free Lie algebra, equipped with an action of $\mu_N$. The Lie algebra $\mathfrak{dmr}_0$ of $\mathrm{DMR}_0$ is a subspace of the Lie algebra $\mathrm{out}^*$, defined by a pair of shuffle relations (Racinet) and containing the Grothendieck-Teichmüller Lie algebra or its analogues(Furusho). We show that the harmonic coproduct may be viewed as an element of a module over $\mathrm{out}^*$, and that $\mathfrak{dmr}_0$ then identifies with the stabilizer Lie algebra of this element. A similar identification concerning $\mathrm{DMR}_0$ enables one to construct a "Betti" version of the harmonic coproduct, and to identify the scheme arising from double shuffle relations as the set of elements of $\mathrm{Out}^*$ taking the harmonic coproduct to its "Betti" version

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