SUPERIOR ALGEBRA 2
Learning outcomes of the course unit
The students will learn the basic definitions, problems, and techniques in the theory of Lie groups. At the end of the course, they will be able to solve basic exercises.
Linear Algebra: canonical Jordan form, diagonalization and triangularization, bilinear, sesquilinear, symmetric, Hermitian, and quadratic forms.
Differential geometry: manifolds and vector fields.
Course contents summary
This course is an elementary introduction to Lie groups and Lie algebras. In the first part, we shall discuss a number of basic notions from differential geometry and the fundamental problems of the theory. In the second part, we shall study the first examples of Lie groups, included matrix Lie groups, and their topology. In the third part, we shall introduce Lie algebras and discuss their role in the theory.
Basic notions of category theory.
Basic notions of differential geometry.
Lie groups: definition and examples. Matrix Lie groups.
Topology of Lie groups.
Lie algebras and their representations.
A certain number of Lecture Notes is freely available online. In particular, we shall refer to the notes by A. Kirillov Jr., A. Savage, and S. Helgason.
The topics of the course will be discussed during the lectures, together with examples, applications, and exercises.
Assessment methods and criteria
Every forth-night one lecture will be focused on exercises, both computational and theoretical. At the end of the course there will be a written exam. A passing grade will give access to a subsequent oral exam, consisting in an interview at the board, during which the student will be asked to discuss, explain, and prove the main results of the course.