A stabilizer interpretation of double shuffle Lie algebras

TL;DR

This paper provides a new interpretation of the double shuffle Lie algebra as a stabilizer in a group action, offering an alternative proof of its Lie algebra structure based on harmonic coproducts.

Contribution

It introduces a stabilizer perspective of the double shuffle Lie algebra, connecting it to harmonic coproducts and group actions, and offers a new proof of its Lie algebra properties.

Findings

Double shuffle Lie algebra identified as a stabilizer group element.

Harmonic coproduct viewed as a module element over a pro-unipotent group.

Alternative proof that the double shuffle Lie algebra is a Lie algebra.

Abstract

According to Racinet's work, the scheme of double shuffle and regularization relations between cyclotomic analogues of multiple zeta values has the structure of a torsor over a pro-unipotent $\mathbb Q$-algebraic group $\sf{DMR}_0$, which is an algebraic subgroup of a pro-unipotent $\mathbb Q$-algebraic group of outer automorphisms of a free Lie algebra. We show that the harmonic (stuffle) coproduct of double shuffle theory may be viewed as an element of a module over the above group, and that $\sf{DMR}_0$ identifies with the stabilizer of this element. We identify the tangent space at origin of $\sf{DMR}_0$ with the stabilizer Lie algebra of the harmonic coproduct, thereby obtaining an alternative proof of Racinet's result stating that this space is a Lie algebra (the double shuffle Lie algebra).