Lie algebras and Geometrical Configurations of Points and Lines

I will be discussing a connection of configurations and simple Lie algebras which we observed during our investigation of the universal dimension formulae of the latter. In the framework of the universal approach (proposed by Vogel, Deligne, et al.), the parametrization of simple Lie algebras is based on the points in the projective plane $\mathbb P^2$. Notably, certain important quantities of the simple Lie algebras are expressed as specific rational functions of homogeneous coordinates of $\mathbb P^2$.Our focus is on the issue of uniqueness regarding these functions. We have discovered that this problem is equivalent to the existence of specific configurations of points and lines. In particular, we have found that valid configurations must be ”colorable” in a specific manner. I will explain how the colorable versions of $(9_3)$ and $(16_3, 12_4)$ configurations have partially resolved the uniqueness problem. Additionally, I will discuss a colorable $(144_3, 36_12)$ configuration that has the potential to completely solve the problem.