Poisson brackets and coaction maps of regularized holonomies of the KZ equation

We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra $\mathrm{g}\mathrm{l}(N,\mathbb{C})$. Our main technical tool is a certain projection of the generalized pentagon equation of \cite{AFR2024}.