Polylogarithmic characterizations of the reduced coaction Lie algebra and the double shuffle Lie algebra, and their relation to the Kashiwara--Vergne Lie algebra

Abstract

The reduced coaction Lie algebra $\mathfrak{rc}_0$ is defined by a skew-symmetric condition together with an algebraic equation involving the reduced coaction, a non-cyclic refinement of the necklace cobracket. The double shuffle Lie algebra $\mathfrak{dmr}_0$ encodes the regularized double shuffle relations in the study of formal multiple zeta values, while the Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2$ arose in the study of the Kashiwara--Vergne conjecture. Our main result is the description of $\mathfrak{dmr}_0$ and $\mathfrak{rc}_0$ by vanishing conditions of different strengths on the same object: the defect of Drinfeld's pentagon equation; $\mathfrak{dmr}_0$ is characterised by the vanishing of the associated two-variable multiple polylogarithms, and $\mathfrak{rc}_0$ by the vanishing of those whose second index has depth $1$. We also establish explicit relations among these three Lie algebras. First, the skew-symmetric elements of $\mathfrak{dmr}_0$ embed into $\mathfrak{rc}_0$. Second, under an additional condition, we construct an injective Lie algebra morphism from $\mathfrak{rc}_0$ to $\mathfrak{krv}_2$.