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
Fundamental theory of Lie groups: basic properties; Lie subgroups; coverings and fundamental groups; homogeneous spaces; classical Lie groups; invariant vector fields and differential forms.
Fundamental theory of Lie algebras: the exponential map; fundamental theorems of Lie theory; representations of Lie algebras; universal enveloping algebras; Poincaré-Birkhoff-Witt theorem; Baker-Campbell-Hausdorff formula.
Semisimple Lie algebras: solvable and nilpotents Lie algebras; Lie's and Engel's Theorems; reductive and semisimple Lie algebras; Cartan's criterion; Killing form and Jordan decomposition; Whitehead's Theorem and Weyl's Theorem of complete reducibility; structure of semisimple Lie algebras; root systems; Weyl groups; classification of irreducible representations; Weyl's character formula.
Representations of compact Lie groups: differential forms and integration theory on Lie groups; representations of compact Lie groups; Peter-Weyl Theorem and applications.
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
The exam is structured in two parts. The first part of the exam will consist of solving 30 exercises chosen from those assigned during the course. The oral exam will consist of a 30-minute seminar for in-depth discussion, accompanied by a brief report, on a topic agreed upon with the instructor. At the end of the presentation, the student will be asked to explain the solution to one of the exercises presented.
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.
