CU Algebraic Lie Theory Seminar

Semisimple Lie Algebras

Shawn Baland, CU

Abstract

Many of the structures encountered in algebraic Lie theory have their origin in the study of finite-dimensional complex semisimple Lie algebras. In this talk, we will examine the existence and conjugacy of the Cartan subalgebras of a finite-dimensional complex Lie algebra. We will then study the module structure induced on a Lie algebra $g$ via the adjoint action of one of its Cartan subalgebras $h$. This will allow us to decompose $g$ in to a direct sum of eigenspaces with with respect to $h$. In the semisimple case, the linear functionals associated with these eigenspaces tell us a lot about a Lie algebra's structure. This information will be used in my second lecture to study the geometry and combinatorics of root systems.

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