A result due to Joyal, Klyachko, and Stanley relates free Lie algebras to partition lattices. We will discuss the precise relationship and interpret the result in terms of the braid hyperplane arrangement. We will then extend the result to arbitrary (finite, real, and central) hyperplane arrangements. Important elements are work of Björner and Wachs on the topology of geometric lattices and a generalization of the classical Dynkin idempotent that we introduce.

This is part of joint work with Swapneel Mahajan in which we extend several aspects of the classical theory of the free Lie algebra to the setting of hyperplane arrangements and develop a corresponding Hopf-Lie dictionary.