MAU34103 Lie algebras and Lie groups
| Module Code | MAU34103 |
|---|---|
| Module Title | Lie algebras and Lie groups |
| Semester taught | Semester 1 |
| ECTS Credits | 5 |
| Module Lecturer | Prof. Sergey Mozgovoy |
| Module Prerequisites |
MAU11101 Linear algebra I and
MAU23206 Calculus on manifolds |
Assessment Details
- This module is examined in a 2-hour examination at the end of Semester 1.
- Continuous assessment contributes 15% towards the overall mark.
- Re-assessment, if needed, consists of 100% exam.
Contact Hours
11 weeks of teaching with 3 lectures per week.
Learning Outcomes
On successful completion of this module, students will be able to
- Define the following concepts: Lie group, Lie algebra, exponential map, homomorphism of Lie algebras, representation of a Lie algebra, subrepresentation, irreducible representation, homomorphism of representations, universal enveloping algebra, Killing form of a Lie algebra, nilpotent Lie algebra, solvable Lie algebra, semisimple and simple Lie algebra, Cartan subalgebra, root system.
- Define and calculate the classical Lie algebras.
- Describe the construction of the irreducible representation of sl2.
- State the fundamental theorem of Lie theory, the PBW theorem, Engel's theorem, Lie's theorem and Cartan's criterion.
- Describe the Jordan-Chevalley decomposition for semisimple Lie algebras.
- Give the root space decomposition and root system of sln.
Module Content
- Lie groups, Lie algebras, examples.
- Lie algebra of a Lie group, exponential map, adjoint representation.
- Universal enveloping algebra, PBW theorem, Casimir element.
- Irreducible representation of sl2(C).
- Nilpotent Lie algebras, Engel's theorem.
- Semisimple Lie algebras, Killing form, Cartan's criterion.
- Complete reducibility (Weyl's theorem).
- Cartan decomposition of a semisimple Lie algebra.
- Irreducible representations of a semisimple Lie algebra.
Recommended Reading
- Introduction to Lie algebras and representation theory by J. Humphreys.
- Lie algebras and Lie groups by J.P. Serre.
- Lie algebras of finite and affine type by R. Carter.