Prerequisites: Basic linear algebra and the elementary theory of rings and bodies.

We will study the theory of Lie algebras and their representations over complex numbers. After familiarizing ourselves with the basic notions of ideal, homomorphism, representation, etc., we will study the structural theory of nilpotent and soluble Lie algebras, and then the Cartan-Killing theory of semisimple Lie algebras, and the classification theorem in terms of root systems and Dynkin diagrams. Finally, we will study the theory of finite-dimensional representations of semisimple Lie algebras, including characters, Weyl’s formula, cohomology, etc. The subject is exclusively algebraic, but connections with Lie group theory and geometry will be mentioned.

References:
HUMPHREYS, J.E. – Introduction to Lie algebras and representation theory (Springer).
KNAPP, A. W. – Lie groups beyond an introduction (Birkhäuser).
KAC, V. G. MIT class notes http://math.mit.edu/classes/18.745/classnotes.html.


Note: This course is offered as a master’s degree. At the PhD, it has additional requirements.

* Standard program. The teacher has the autonomy to make any changes.