CU Algebraic Lie Theory Seminar

The combinatorics of root systems

Shawn Baland, CU

Abstract

The roots of a finite-dimensional complex semisimple Lie algebra span a Euclidean space with respect to the Lie algebra's Killing form. By assigning a total ordering to this space, we can partition the roots into what are called positive and negative root systems. In this talk, we will examine a subset of the positive system that forms a basis of the Euclidean space. The relative lengths and angles between the roots in this subset will be encoded into a matrix called the Cartan matrix. We will then relate the Cartan matrix to a graph called the Dynkin diagram, and it will be shown that there are only a limited number of possible connected Dynkin diagrams. Finally, we will discus the correspondence between these diagrams and the finite-dimensional complex simple Lie algebras.

A copy of the slides of the talk is available here.