Polylogarithmic functions (polylogs) in n variables can be viewed as elements of (U\mathfrak{p}_{m})^*, 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})^* 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.
