Module Catalogue 2024/25

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

To introduce the concept of a Lie group and its Lie algebra, study the interplay between their analytical, topological and algebraic properties and develop the representation theory of Lie algebras.

Module summary
This course is an introduction to Lie theory, the study of continuous symmetries. Lie's original vision at the end of the 19th century was to develop a theory of symmetries of differential equations (described by what we call nowadays Lie groups). These Lie groups play a central role in many areas of mathematics and physics. In this course we will focus on matrix Lie groups which are easier to study, but still contain the most important examples such as the general linear, orthogonal and symplectic groups. Any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. The Lie algebra is an algebraic linearization of the Lie group and usually easier to work with. There is a close correspondence between the Lie algebra and the Lie group which we will develop in this course. This correspondence allows one to study the structure of Lie groups in terms of Lie algebras. Often times Lie groups and Lie algebras are studied via their representations, that is, the way they can act (linearly) on vector spaces. We will study the representation theory of classical matrix Lie groups and semisimple Lie algebras and apply it in many examples.

Outline Of Syllabus

Matrix Lie groups. The matrix exponential. The Lie algebra of a Lie group. Representations of Lie groups and Lie algebras. The exponential map and its properties. The Baker-Campbell-Hausdorff formula. Lie's theorems. Representations of sl(2,C). Representations of the classical semisimple Lie algebras of type ABCD. Complete reducibility. Root space decompositions and highest weight theory for classical semisimple Lie algebras

Intended Knowledge Outcomes

Students will be able to demonstrate an understanding of the structure and representation theory of matrix Lie groups and their associated Lie algebras. They will have an understanding of the representation theory of semisimple Lie algebras, including the appearance in other areas such as geometry or physics.

Intended Skill Outcomes

Upon succesful completion of the module students will be able to: translate between properties
of Lie groups and Lie algebras using Lie’s theorems. They will know how to decide whether a Lie algebra is semisimple. They will be able to construct representations of semisimple Lie algebras, describe these explicitly in types ABCD and be able compute numerical invariants such as
the dimension.

Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised
edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Tutorials (within lectures) are used to discuss the course material, identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

The written examination is split into two parts due to the wish of the students.

General Notes

N/A