On successful completion of this module, students will be able to

Give the definitions of: Lie group, Lie algebra, exponential map, homomorphism of Lie algebras, representation of a Lie algebra, subrepresentation, irreducible representation, homomorphism of representations, universal enveloping algebra, the Killing form of a Lie algebra, nilpotent Lie algebra, solvable Lie algebra, semisimple and simple Lie algebra, Cartan subalgebra, root system.

Give the definitions of and calculate with the classical Lie algebras.

Describe the construction of the irreducible repesentation of sl2.

State the fundamental theorem of Lie theory, 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.

Irreducible representations of a semisimple Lie algebra.

Module Prerequisite

MA1111, (MA2314 & MA2322 recommended)

Literature:

Humphreys, Introduction to Lie algebras and representation theory.

Serre, Lie algebras and Lie groups.

Carter, Lie algebras of finite and affine type.

Assessment Detail

This module will be examined in a 2 hour examination in Trinity term.
Continuous assessment
will contribute 15% to the final grade for the module at the annual
examination session. Supplementals if required will be assessed 100% exam.