Unit name | Lie groups, Lie algebras and their representations |
---|---|

Unit code | MATHM0012 |

Credit points | 10 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 1A (weeks 1 - 6) |

Unit director | Professor. Robbins |

Open unit status | Not open |

Pre-requisites |
MATH11005 (Linear Algebra and Geometry), MATH11006 (Analysis 1), MATH11007 (Calculus 1), MATH 20901 (Multivariable calculus) (or equivalently, Calculus 2). 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. |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

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.

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.

Lie groups also appear in the syllabus of the proposed new unit 'Topics in Modern Geometry'. However, 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. A student who takes both units will benefit from seeing distinct parts of the subject seen from different perspectives.

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; formulate and apply the Cartan criterion for determining semisimplicity; 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 semisimple Lie algebras, including the Killing form, adjoint representation, Cartan subalgebra, and weights and roots;
- explain and construct Dynkin diagrams;
- give a complete classification of finite-dimensional semisimple Lie algebras, including theexceptional 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;
- define tensor product representations, and decompose tensor products into irreducible components;
- explain and apply some aspects of the representation theory of noncompact semisemple Lie groups in specific examples;
- 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.

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

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

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)