Equivalence of krv1 and a polylogarithmic identity for skew-symmetric Lie series

Establish that for any skew-symmetric Lie series ψ ∈ 𝔣𝔯_k(x_0,x_1) with c_{x_0}(ψ)=c_{x_1}(ψ)=0, the Kashiwara–Vergne equation krv1, namely [x_1, ψ(−x_0−x_1, x_1)] + [x_0, ψ(−x_0−x_1, x_0)] = 0, is equivalent to the two-variable polylogarithm identity l^{y,x}_{(a_1,…,a_m),(b_1,…,b_n)}(ψ_{451} + ψ_{123}) = l^{y,x}_{(a_1,…,a_m,b_1),(b_2,…,b_n)}(ψ_{451} + ψ_{123}) for all index pairs (a_1,…,a_m),(b_1,…,b_n) ≠ (1,…,1),(1,…,1), where ψ_{ijk} := ψ(x_{ij}, x_{jk}) in the spherical braid Lie algebra 𝔭_5 and l^{y,x}_{·,·} denotes the bar elements associated to multiple polylogarithms on 𝓜_{0,5}.

Background

The paper studies the reduced coaction Lie algebra 𝔯𝔠0 and its connections to the double shuffle Lie algebra 𝔡𝔪𝔯_0 and the Kashiwara–Vergne Lie algebra 𝔨𝔯𝔳_2. The authors prove that skew-symmetric elements of 𝔡𝔪𝔯_0 inject into 𝔯𝔠_0, and that 𝔯𝔠_0 together with the krv1 equation maps to 𝔨𝔯𝔳_2. A central technical tool is the polylogarithmic description of defects of the pentagon equation, expressed via bar elements l^{y,x}{\mathbf{a},\mathbf{b}} on 𝓜_{0,5}.

The conjecture asks for a precise equivalence between the algebraic commutator condition krv1 and a family of polylogarithmic identities involving the combination ψ{451}+ψ{123}. If true, it would imply that 𝔡𝔪𝔯_0 with the skew-symmetric condition coincides with 𝔨𝔯𝔳_2 under skew symmetry, thereby unifying two major structures connected to multiple zeta values and the Kashiwara–Vergne problem.