The kernel of formal polylogarithms

Polylogarithmic functions (polylogs) in $n$ variables can be viewed as elements of $(U{\mathfrak{p}}_m{)}^{\ast }$, the dual of the universal enveloping algebra of the Lie algebra ${\mathfrak{p}}_m$ of infinitesimal spherical pure braids with $m=n+3$ strands. Polylogs with $m=4,5$ are used in the theory relating double shuffle relations and Drinfeld associators \cite{furusho_double_2011}. We give explicit formulas for elements of $(U{\mathfrak{p}}_m{)}^{\ast }$ representing polylogs, and compute the left ideal $J_m\subset U{\mathfrak{p}}_m$ given by their joint kernel. We introduce Lie subalgebras ${\mathfrak{k}}_m={\mathfrak{p}}_m\cap J_m$, and we compute them for $m=4,5$.