Lie Algebras (0590312) 10 credits

Introduction to Lie Algebras and their Representation Theory

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Aims

To introduce students to finite-dimensional Lie algebras over the complex field, to give elements of their representation theory, and to show how Lie algebras appear in other branches of Mathematics, including Differential Geometry and Mathematical Physics.

Learning Objectives

By the end of the module students should:

• learn basic properties of solvable and semisimple Lie algebras,
• know the description of irreducible finite-dimensional representations of the Lie algebra sl₂ ,
• be aware of the classification of the finite-dimensional simple Lie algebras.

• Matrices, Linear Algebra, Geometry (0500007)
• Linear Algebra (0590014)
• Groups and Rings (0590019)

1. Basic examples of finite-dimensional Lie algebras.
2. Lie algebras in dimensions one, two and three.
3. Finite-dimensional representations of the Lie algebra sl₂ .
4. Basic properties of solvable Lie algebras.
5. Lie theorem for solvable Lie algebras.
6. Basic properties of semisimple Lie algebras.
7. Classification of simple Lie algebras.

1. W Fulton and Harris, Representation theory: a first course, Springer, 1991 (S 2.86 FUL) **
2. J E Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972/1978 (S 2.89 HUM) ***
3. N Jacobson, Lie algebras, Interscience, 1962 (S 2.89 JAC) **
4. H Samelson, Notes on Lie algebras, Van Nostrand, 1969 (S 2.89 SAM) ***
5. J-P Serre, Lie algebras and Lie groups, W A Benjamin, 1965 (QUARTO S 2.86 SER) *
6. J-P Serre, Complex semisimple Lie algebras, Springer, 1987 (S 2.89 SER) **
7. I Stewart, Lie algebras, Springer, 1970 (North Room QUARTO S 0.4 LEC ) *

• 2 lectures per week
• 1 problems class per week

Term taught

Autumn

Assessment

• Examination in Week 1 of Spring Term (90%)
• Coursework (10%)

Elective information

Not available

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Revised 5 March 2009