Learning the basic notions and concepts both of the theory of Lie groups and the theory of representations.
Course Prerequisites
The knowledge of the basic mathematical tools, typically learned during the BSc courses, is required. It is advisable that the student has studied or is studying differential geometry.
Teaching Methods
On account of the theoretical and mathematical nature of the course, all classes will be given at the blackboard and all topics of the programme will be discussed.
Assessment Methods
The student has to give an oral exam aimed at verifying that he/she has learned the topics presented during the lectures. Special emphasis will be given to the verification of the ability of the student to apply and rigorously present all concepts learned.
Texts
F. Warner "Foundations of differentiable manifolds and Lie groups" (1990) 3ed. Springer-Verlag. J. Lee "Introduction to smooth Manifolds"(2003) 2ed. Springer. A. W. Knapp "Lie groups: Beyond an introduction" (2005) Birkhäuser A. O. Barut, R. Raczka "Theory of Group representations and applications" (1986) World Scientific.
Contents
In the first part of the course, we present the basic structural properties of the theory of groups (subgroups, quotients, actions, extensions, presentations). Then, we will present the basic results of the theory of Lie groups and Lie algebras, with special emphasis on the geometrical and topological aspects. Subsequently, we will study the fundamentals of representation theory. In particular, we will focus on the basic results of the representation theory of finite and compact groups, the Frobenius character theory, Haar measure and unimodularity, tensor representations.
