Lie Algebras

Syllabus Details

• Unit I: Definition of Lie algebras, classical examples, ideals, standard isomorphism theorems, nilpotent Lie algebras, solvable Lie algebras, simple Lie algebras, Engel’s theorem.
• Unit II: Lie’s theorem, Jordan decomposition, Cartan’s criterion for solvability, Cartan-Killing form, semisimplicity, SL(2) representations.
• Unit III: Total and and Cartan subalgebras leading to root systems.
• Unit IV: Learn definition of root systems, basics, Weyl group., their properties.
• Unit V: Study of simple root systems, Weyl group, simple root systems, Dynkin diagrams, classification of simple root systems.
• Unit VI: Universal enveloping algebra, statement of PBW theo- rems, roots and weights calculation in classical set up, statement of Serre’s theorem, definition and basic properties of Verma modules, statement of classification of finite dimensional representations of simple Lie algebras.

Course Learning Objective

1. Learn basics of Lie algebras, solvable and nilpotent Lie algebras.
2. Learn semisimple Lie algebras, Cartan’s criterion, connection between solvable and semi simple Lie algebras, classification of SL(2) modules, Weyl’s theorem on complete reducibility.
3. Learn notion of total and Cartan subalgebras, using Cartan subalgebras slowly lead to root systems.
4. Learn basics of root systems, Weyl group combinatorics.
5. Study Classification of root systems leading to classification of simple Lie algebras.
6. Universal enveloping algebra definition, statement of Serre’s theorem, basics of Verma modules and statement of classification of finite dimensional irreducible representations of semisimple Lie algebras.

Recommended Books

1. Humphreys, J., Introduction to Lie Algebras and Representation Theory, GTM 9, Springer-Verlag, 1972.
2. Serre, Jean-Pierre, Complex Semisimple Lie Algebras, Springer Monographs, 2001.