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

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