Math 650 — Spring 2002 Lie Groups and Lie Algebras

This is an introduction to the theory of Lie groups and algebras and their linear representations — a fundamental part of many branches of Mathematics (algebra, differential and algebraic geometry, topology, harmonic analysis, differential equations...) and an important tool in modern Physics (elementary particles, gauge theory, strings...). Only a basic knowledge of mathematical analysis and linear algebra is required. Elements of theory of differentiable manifolds will be introduced as needed. I shall try to avoid generalities and technicalities and to emphasise ideas illustrated on concrete examples.

The following topics will be covered.

1. Symmetries in Physics and Lie groups. Applications to the elementary particles theory.
2. The groups of real and complex matrices and their classical subgroups. The corresponding Lie algebras. Exponential mapping.
3. General concept of a Lie algebra. Construction of the corresponding Lie group via the Campbell-Hausdorff formula.
4. Algebras of differential operators and groups of transformations of a differentiable manifold. Systems of linear PDE's of the first order.
5. Universal Lie algebra. Its center and invariant differential operators.
6. Invariant tensor fields on homogeneous spaces. The Laplace-Beltrami operators.
7. Geometric integration theory.
8. Compact Lie algebras.
9. Solvable, simple and semi-simple Lie algebras
10. Structure of semisimple Lie algebras. Root systems. Simple roots.
11. The Weyl and Coxeter groups.
12. Linear representations of semisimple groups. Description of an irreducible representation by the highest weight. Casimir element and weight multiplicities. A tensor construction. Weyl's character formula.

Bibliography

1. J.-P. Serre, Lie Algebras and Lie Groups, Lectures given at Harvard University, Benjamin, New York-Amsterdam, 1965; Springer, New York, 1992.
2. J.-P. Serre, Complex Semisimple Lie Algebras, Springer, New York, 1987.
3. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Third Printing, Springer, New York, 1960.
4. E. B. Dynkin, The structure of semi-simple Lie algebras, Amer. Math. Society Translations, Number 17, New York, 1950.
5. E. B. Dynkin, Survey of the basic concepts and facts in the theory of linear representations of semisimple Lie algebras, Supplement in: E. B. Dynkin, Maximal subgroups of the classical groups, Amer. Math. Society Translations, Series 2, Providence, R. I., 1957.
6. R. N. Cahn, Semi-simple Lie Algebras and their Representations, Benjamin/Cummings, Menlo Park, CA, 1984.
7. S. Sternberg, Group Theory and Physics, Cambridge University Press, Cambridge, 1994.