# Mathematics / Applied Mathematics MSc

# 5M: Lie Groups, Lie Algebras and their Representations MATHS5049

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Either Semester 1 or Semester 2**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

The aim of this course is to explore the concept of symmetry in mathematics, meaning the study of Lie groups or related structures and of their actions on other mathematical objects.

#### Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

#### Requirements of Entry

Mandatory Entry Requirements

4H Topics in Algebra.

Recommended Entry Requirements

4H Differential Geometry

#### Assessment

Assessment

100% Final Exam

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

#### Course Aims

The aim of this course is to explore the concept of symmetry in mathematics, meaning the study of Lie groups or related structures (e.g. Lie algebras, Hopf algebras, monoidal categories) and of their actions on other mathematical objects. The theory has a wide range of applications especially in algebra, geometry, topology and physics, some of which will also be discussed.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

• Demonstrate knowledge of the central definitions and facts of selected topics in the theory of Lie groups and Lie algebras and use these to solve problems of a numerical or logical nature. Two topics will be taken from the following:

1. Lie groups and their Lie algebras (basic definitions, examples and applications, exponential map, Baker-Campbell-Hausdorff formula)

2. Lie groups and geometry (calculus on manifolds, invariant forms, homogeneous and symmetric spaces, symmetries of differential equations)

3. Structure theory of Lie algebras (Engel's theorem, Lie's theorem, Cartan's criterion, structure and classification of complex simple Lie algebras)

4. Representation theory (examples and applications, representations of simple Lie algebras, Weyl character formula, Lie algebra cohomology)

• Generalise parts of the theory to overarching structures (e.g. Hopf algebras

and monoidal categories).

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.