The kernel of formal polylogarithms

TL;DR

This paper provides explicit formulas for polylogarithmic functions in multiple variables, explores their algebraic structures, and computes associated Lie subalgebras, advancing understanding of their role in number theory and algebraic geometry.

Contribution

It introduces explicit formulas for polylogarithms in the dual of the universal enveloping algebra and computes the related Lie subalgebras for specific cases.

Findings

Explicit formulas for polylogarithms in $(Urak{p}_m)^*$.

Identification of the joint kernel ideal $J_m$ in $Urak{p}_m$.

Computation of the Lie subalgebras $rak{k}_m$ for $m=4,5$.

Abstract

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 $m=4, 5$.