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 via the adjoint action of one of its Cartan subalgebras . This will allow us to decompose in to a direct sum of eigenspaces with with respect to . 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.
Semisimple Lie Algebras