MATH 7410 - Lie Combinatorics and Hyperplane Arrangements

Marcelo Aguiar, fall 2016.

The goal of this course is to study various algebraic, geometric and combinatorial aspects of hyperplane arrangements. We will extend several aspects of the classical theory of the free Lie algebra to a new theory relative to a hyperplane arrangement. The classical case is obtained by specializing to the braid arrangement. The main objects of study are the monoid of flats and the monoid of faces of the arrangement, and their representation theory. We will introduce the notions of Lie element, noncommutative Mobius function, Eulerian idempotent, and Dynkin idempotent, among others. Time permitting, we will discuss how the notions of operad, Hopf algebra, Lie algebra, and others, can be extended to the setting of hyperplane arrangements.

Basic knowledge of ring theory and of the combinatorics of hyperplane arrangements will be useful, although I will attempt to cover most of the prerequisites.