# Lie Groups, Lie Algebras and Their Representations

In 2018/19 this unit will not be run as a TCC unit. The unit will run as a regular M Level unit, some of the content listed below will therefore be removed from the curriculum. We will update this information as soon as possible.

## Unit aims

The aims of this unit are to introduce the principal elements of semisimple Lie groups, Lie algebras and their representations, for which there is a relatively complete and self-contained theory. The course will develop conceptual understanding as well as facility with calculation. By treating semisimple Lie groups as sets of finite-dimensional matrices (the alternative, more abstract point of view is to treat them as differentiable manifolds), the unit will be made accessible to a students with a broad range of backgrounds.

## Unit description

Lie groups and Lie algebras embody the mathematical theory of symmetry (specifically, continuous symmetry). A central discipline in its own right, the subject also cuts across many areas of mathematics and its applications, including geometry, partial differential equations, topology and quantum physics. This unit will concentrate on finite-dimensional semisimple Lie groups and Lie algebras and their representations, for which there exists a rather complete and self-contained theory. Applications will be discussed. Students will be expected to have attained a degree of mathematical maturity and facility at least to the standard of a beginning level 7 student.

## Relation to other units

This unit complements the following units:

- Group Theory (MATH33300) and Representation Theory (MATH M4600) deal mainly with finite groups, but there are some strong parallels with the theory of compact Lie groups.

- Differentiable Manifolds 3 (MATH 32900) and Differentiable Manifolds 34 (MATH 2900) provide the framework for the geometry of Lie groups. In turn, Lie groups, and spaces on which they act, constitute an important class of differentiable manifolds

- Mechanics 2, Mechanics 23 (MATH 31910), Quantum Mechanics (MATH 35500), Quantum Chaos (MATH 5700), Quantum Information Theory (MATH 5600). As the basis for the mathematical theory of continuous symmetries, Lie groups and Lie algebras play a prominent role in physics, particularly in classical mechanics and especially in quantum mechanics.

- Algebraic Topology (MATH 1200). Many Lie groups have interesting topological properties, and a number of central results in algebraic topology concern Lie groups.

- Random Matrix Theory (MATH 33720) concerns the statistical distribution of eigenvalues of sets of matrices averaged over the action of certain Lie groups with respect to Haar measure.

- Lie groups may also appear as a topic in 'Topics in Modern Geometry'. The point of view as well as the specific content in that unit will be independent of and complementary to the material covered in this one.

## Learning objectives

A student successfully completing this unit will be able to:

- state the definition of matrix Lie groups and Lie algebras, explain the connections between them, and describe their relationship to symmetry;
- state the definition of semisimple Lie groups and Lie algebras;
- delineate the principal examples; construct examples of non-semisimple Lie algebras;
- formulate the definitions of representation, irreducible representation and complete reducibility;
- prove and apply Schur’s Lemma, formulate and apply a procedure for reducing a given representation into irreducible components;
- explain the principal elements of the representation theory of finite-dimensional compact Lie algebras, including the adjoint representation, Cartan subalgebra, and weights and roots;
- explain and construct Dynkin diagrams;
- give a complete classification of finite-dimensional compact Lie algebras, including the exceptional Lie algebras, and prove the principal theorems required for the classification;
- define Haar measure; calculate Haar measure and evaluate integrals over groups in specific examples;
- explain and apply the principal elements of the representation theory of compact semisimple Lie groups, including unitarity, orthogonality and completeness, and prove the principal theorems;
- appreciate how the subject relates to some other areas of mathematics and physics, including, for example, differential geometry, partial differential equations, and/or quantum mechanics and quantum information theory;
- apply results from the unit to problems in these areas.

## Transferable skills

Mathematical skills: Knowledge of Lie groups, Lie algebras, representation theory, in particular for compact groups; knowledge of principal examples; geometrical and algebraic reasoning.

General skills: Problem solving and logical analysis; Assimilation and use of complex and novel ideas

## Syllabus

- Review of linear algebra. Tensor products. Definition of matrix Lie groups. Examples. Lie algebras.
- Haar measure.
- Representation theory for compact matrix Lie groups: Representations. Equivalence. Irreducibility and complete reducibility. Schur's Lemma. Peter-Weyl theorem.
- Compact simple Lie algebras. Killing form, Cartan subalgebra, roots and weights. Dynkin diagrams. Classification.

## Reading and References

Typset lecture notes will be provided. The following texts cover different parts of the unit in greater detail:

H Georgi, Lie Algebras and Particle Physics, 2nd edition (1999)

K Erdmann and MJ Wildon, Introduction to Lie Algebras, Springer (2004)

B Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer-Verlag (2004)

JJ Duistermaat and JAC Kolk, Lie groups, Springer-Verlag (2000)

**Unit code:** MATHM0012

**Level of study:** M/7

**Credit points:** 10

**Teaching block (weeks):** 1 (1-6)

**Lecturer: **Dr Nugzar Suramlishvili

## Pre-requisites

MATH11005 Linear Algebra and Geometry, MATH10003 Analysis 1A and MATH10006 Analysis 1B, MATH11007 Calculus 1, MATH20901 Multivariable Calculus, MATH21100 Linear Algebra 2.

## Co-requisites

None

## Methods of teaching

The unit will be delivered through lectures. Lecture notes will be provided. Problem sheets will be assigned and marked, and solutions distributed.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 100% from a 1 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.