Module description
Syllabus
Basic definitions and examples of Lie groups and Lie algebras. Matrix Lie groups, their Lie algebras; the exponential map, Baker-Campbell-Hausdorff formula. Representations of Lie algebras, sub-representations, Schur's Lemma, tensor products. Root systems, Cartan-Weyl basis, classification of simple Lie algebras (perhaps with some of the proofs being left out.)
Prerequisites
Before taking this module you are advised to have knowledge and experience of the topics covered in the syllabus for 4CCM121A/5CCM121B Introduction to Algebra, 5CCM226A/6CCM226B Metric Spaces & Topology
Assessment details
2 hr written examination.
Semester 1 only students will be set an alternative assessment in lieu of in-person exams in January.
Full year students will complete the standard assessment.
Educational aims & objectives
This course gives an introduction to the theory of Lie groups, Lie algebras and their representations, structures which arise frequently in mathematics and physics.
Lie groups are, roughly speaking, groups with continuous parameters, the rotation group being a typical example. Lie algebras can be introduced as vector spaces (with extra structure) generated by group elements that are infinitesimally close to the identity. The properties of Lie algebras, which determine those of the Lie group to a large extent, can be studied with methods from linear algebra, and one can even address the question of a complete classification.
Teaching pattern
Two hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists