SUPERIOR ALGEBRA 1
cod. 1010205

Learning objectives

The students will learn the basic definitions, problems, and techniques in the theory of Lie groups.

Prerequisites

Algebra (groups, rings, fields); Linear Algebra; Differential Geometry.

Course unit content

This course is an introduction to Lie groups and Lie algebras. In the first part, we shall discuss several 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. The three main goals of the course are the Peter-Weyl theorem, the Weyl character formula, and the classification theorem of simple complex Lie algebras.

Full programme

Bibliography

A high number of Lecture Notes on Lie groups and Lie algebras is freely available online. We shall refer to the notes by P. Etingof and A. Kirillov Jr.:
[E] P. Etingof, Lie groups and Lie algebras
https://arxiv.org/abs/2201.09397
[K] A. Kirillov Jr., An Introduction to Lie groups and Lie algebra, Cambridge University Press.

Teaching methods

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 an exam in two parts. The first one will be a one-hour long written exam. The second one will be at the board, where the student will be asked to discuss, explain, and prove the main results of the course.

Other information

NA