Learning objectives
The students will learn the basic definitions, problems, and techniques in the theory of Lie groups and Lie algebras.
Prerequisites
Algebra (groups, rings, fields); Linear Algebra; Differential Geometry.
Course unit content
Lie groups are fundamental objects in Mathematics and Physics.
This course is an introduction to matrix Lie groups and their Lie algebras. In the first part, we shall discuss several basic notions and the fundamental problems of the theory. In the second part, we shall study the structure of Lie groups and their properties. 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
The main reference for the course is the book "Lie Groups, Lie Algebras, and Representations" by Brian Hall. Several other Lecture Notes on Lie groups and Lie algebras are freely available online. In particular, we point out 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 algebras, Cambridge University Press.
Teaching methods
The topics of the course will be discussed during the lectures, together with examples, applications, and exercises. Attendance is highly recommended.
Assessment methods and criteria
At the end of the course there will be an exam in two parts. The first one will be a written exam with both computational and theoretical exercises. Every forth-night one lecture will be focused on exercises, aimed to assist the students in view of the final exam. The second part of the exam will consist in a short seminar. The list of potential topics will be divided in three categories: advanced results in Lie theory: interplay between Lie theory and particle physics; history of Lie theory.
Other information
The courses of “Algebra Superiore 1” and “Algebra Superiore 2” are completely independent. However, they are complementary. The attendance of both courses may be beneficial, and it is strongly recommended.
