The kernel of formal polylogarithms
Anton Alekseev, Megan Howarth, Florian Naef, Muze Ren, Pavol Ševera
TL;DR
The paper develops an algebraic framework for formal polylogarithms by identifying polylogarithms as elements of $(U\mathfrak{p}_m)^*$, where $\mathfrak{p}_m$ is the Lie algebra of infinitesimal spherical pure braids with $m=n+3$ strands. It constructs the joint kernel $J_m$ of formal polylogs as a left ideal in $U\mathfrak{p}_m$, giving an explicit description $J_m = U\mathfrak{p}_m\,s(\mathfrak{p}_{m-1}) \oplus I_m$ and a complement $C_m$, together with a canonical basis for $\mathcal{L}_m$ and its annihilator $\mathcal{L}_m^\perp$. The authors define Lie subalgebras $\mathfrak{k}_m=\mathfrak{p}_m\cap J_m$ and compute them for $m=4,5$, obtaining $\mathfrak{k}_4=\mathbb{C}X_{\beta\omega}$ and $\mathfrak{k}_5=\mathbb{C}X_{\beta\omega}\oplus[\lambda,\lambda]\oplus s(\mathfrak{p}_4)$ with $\lambda=[ [X_{1\beta}],[X_{2\beta}] ]$. By employing quadratic duality and iterated integrals, the paper links formal polylogs to Drinfeld associators and double shuffle relations, providing explicit descriptions of $\mathfrak{pl}_m=\pi(\mathfrak{p}_m)$ and revealing detailed free-Lie-algebra structures that govern polylogarithmic relations. These results advance the understanding of the algebraic underpinnings of polylogarithms and their role in related structures such as Kashiwara–Vergne theory and associator formalisms.
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 $m=4, 5$.