Synopsis for C3.4b: Lie Groups

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code tbc).

Recommended Prerequisites

Part A Group Theory, Topology and Multivariable Calculus.

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, and links to both algebra and analysis. Lie groups are groups which are simultaneously manifolds, so that the notion of differentiability makes sense, and the group multiplication and inverse maps are differentiable. However this course introduces the theory in a more concrete way via groups of matrices, in order to minimise the prerequisites.

Learning Outcomes

Students will have learnt the basic theory of topological matrix groups and their representations. This will include a firm understanding of root systems and their role for representations.

Synopsis

The exponential map for matrices, Ad and ad, the Campbell–Baker–Hausdorff series.

Linear Groups, their Lie algebras and the Lie correspondence. Homomorphisms and coverings of linear groups. Examples including , and .

The compact and complex classical Lie groups. Cartan subgroups, Weyl groups, weights, roots, reflections.

Informal discussion of Lie groups as manifolds with differentiable group structures; quotients of Lie groups by closed subgroups.

Bi-invariant integration on a compact group (statement of existence and basic properties only). Representations of compact Lie groups. Tensor products of representations. Complete reducibility, Schur's lemma. Characters, orthogonality relations.

Statements of Weyl's character formula, the theorem of the highest weight and the Borel–Weil theorem, with proofs for only.

1. W. Rossmann, Lie Groups: An Introduction through Linear Groups, (Oxford, 2002), Chapters 1–3 and 6.
2. A. Baker, Matrix Groups: An Introduction to Lie Group Theory, (Springer Undergraduate Mathematics Series).

Further Reading

1. J. F. Adams, Lectures on Lie Groups (University of Chicago Press, 1982).
2. R. Carter, G. Segal and I. MacDonald, Lectures on Lie Groups and Lie Algebras (LMS Student Texts, Cambridge, 1995).
3. J. F. Price, Lie Groups and Compact Groups (LMS Lecture Notes 25, Cambridge, 1977).