Section: Course Materials | C3.5 Lie Groups (2025-26)

Lecturer: Profile: Pierrick Bousseau

Course information

General prerequisites:

ASO: Group Theory, A5: Topology and ASO: Introduction to Manifolds are all useful but not essential. It would be desirable to have seen notions of derivative of maps from \(R^n\) to \(R^m\), inverse and implicit function theorems, and submanifolds of \(R^n\). Acquaintance with the notion of an abstract manifold would be helpful but not really necessary.

Course term: Hilary

Course lecture information: 16 lectures.

Course weight: 1

Course level: M

Assessment type: Written Examination

Course overview:

The theory of Lie Groups is one of the most beautiful developments of pure mathematics in the twentieth century, with many applications to geometry, theoretical physics and mechanics. The subject is an interplay between geometry, analysis and algebra. Lie groups are groups which are simultaneously manifolds, that is geometric objects where the notion of differentiability makes sense, and the group multiplication and inversion are differentiable maps. The majority of examples of Lie groups are the familiar groups of matrices. The course does not require knowledge of differential geometry: the basic tools needed will be covered within the course.

Learning outcomes:

Students will have learnt the fundamental relationship between a Lie group and its Lie algebra, and the basics of representation theory for compact Lie groups. This will include a firm understanding of maximal tori and the Weyl group, and their role for representations.

Course synopsis:

Brief introduction to manifolds. Classical Lie groups. Left-invariant vector fields, Lie algebra of a Lie group. One-parameter subgroups, exponential map. Homomorphisms of Lie groups and Lie algebras. Ad and ad. Compact connected abelian Lie groups are tori. The Campbell-Baker-Hausdorff series (statement only).

Lie subgroups. Definition of embedded submanifolds. A subgroup is an embedded Lie subgroup if and only if it is closed. Continuous homomorphisms of Lie groups are smooth. Correspondence between Lie subalgebras and Lie subgroups (proved assuming the Frobenius theorem). Correspondence between Lie group homomorphisms and Lie algebra homomorphisms. Ado's theorem (statement only), Lie's third theorem.

Basics of representation theory: sums and tensor products of representations, irreducibility, Schur's lemma. Compact Lie groups: left-invariant integration, complete reducibility. Representations of the circle and of tori. Characters, orthogonality relations. Peter-Weyl theorem (statement only).

Maximal tori. Roots. Conjugates of a maximal torus cover a compact connected Lie group. Weyl group. Reflections. Weyl group of \(U(n)\). Representations of a compact connected Lie group are the Weyl-invariant representations of a maximal torus (proof of inclusion only). Representation ring of maximal tori and \(U(n)\).

Killing form. Remarks about the classification of compact Lie groups.

Section outline

- Select activity C3.5 Lie Groups Lecture Notes HT 2026
  
  C3.5 Lie Groups Lecture Notes HT 2026 File
  
  These are lecture notes from Nigel Hitchin’s 2015 lectures. In addition, handwritten notes will be provided here following each lecture.
- Select activity Sheet 1 C3.5 Lie Groups HT2026
  
  Sheet 1 C3.5 Lie Groups HT2026 Assignment
  
  Due: Wednesday, 4 February 2026, 11:00 AM
  
  The questions on this sheet are sorted into three sections.
  Section A is for practice and not to be handed in for marking. Solutions will be made available.
  Section B is to be handed in for marking.
  Section C is optional and not to be handed in for marking; you may wish to look at this later.
- Select activity Sheet 2 C3.5 Lie Groups HT2026
  
  Sheet 2 C3.5 Lie Groups HT2026 Assignment
  
  None of the questions should be submitted for marking. Solutions for Sections A and B will be provided at a later stage.
- Select activity Sheet 3 C3.5 Lie Groups HT2026
  
  Sheet 3 C3.5 Lie Groups HT2026 Assignment
  
  Due: Wednesday, 4 March 2026, 11:00 AM
  
  The questions on this sheet are sorted into three sections.
  Section A is for practice and not to be handed in for marking. Solutions will be made available.
  Section B is to be handed in for marking.
  Section C is optional and not to be handed in for marking; you may wish to look at this later.
- Select activity Sheet 4 C3.5 Lie Groups HT2026
  
  Sheet 4 C3.5 Lie Groups HT2026 Assignment
  
  None of the questions should be submitted for marking. Solutions for Sections A and B will be provided at a later stage.

Main content blocks

Section outline