Unit information: Lie groups, Lie algebras and their representations in 2013/14

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 1 (weeks 1 - 12)
Unit director	Professor. Robbins
Open unit status	Not open
Pre-requisites	MATH11005 (Linear Algebra and Geometry), MATH11006 (Analysis 1), MATH11007 (Calculus 1), MATH20900 (Calculus 2)
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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.

Intended Learning Outcomes

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 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; 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.

Teaching Information

The unit will be delivered through lectures. The lectures will be transmitted over the internet as part of the Taught Course Centre (TCC). The TCC is a consortium of five mathematics departments, including Bath, Bristol, Imperial College, Oxford and Warwick.

Assessment Information

Formative homework exercises will be assigned throughout the unit.

The final assessment mark will be based on a 1½-hour written examination (100%).

Reading and References

• H Georgi, Lie Algebras and Particle Physics, 2nd edition (1999)
• B Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer-Verlag (2004)
• JJ Duistermaat and JAC Kolk, Lie groups, Springer-Verlag (2000)