Coursework will count for 15% of the final grade.
Lie algebras should be thought of as the infinitesimal analogue of groups. Lie theory is an important and very active branch of mathematics with many links to other areas: geometry, representation theory, mathematical physics....
The only prerequisites for this course are linear algebra and some knowledge about groups and rings. However, it would be good to get some idea about the natural companion to Lie algebras: Lie groups. For this one needs some idea of what a smooth manifold is: smooth functions on a manifold, the tangent space at a point, a vector field on a manifold. Everything about Lie groups is NON-EXAMINABLE and only to help your own understanding.
The pdf document Lie_algebras_are_infinitesimal_groups
below explains why Lie algebras should be thought of as infinitesimal groups. It is probably best to ignore all formalities and just think of $\delta$ and $\eta$ as ordinary numbers and calculate upto the first order.
The formal definitions of a Lie group and of an algebraic group, the definition of the Lie algebra of a Lie group, and a small guide to the literature can be found in the pdf document further_notes
below. Of course this material is much more advanced and you will probably have to do some further reading before you really understand it.
On successful completion of this module, students will be able to:
- Give the definitions of: Lie algebra, homomorphism of Lie algebras, subalgebra, ideal, derivation, centre, representation of a Lie algebra, submodule, irreducible module, homomorphism of g-modules, the Killing form of a Lie algebra and the trace form of a classical Lie algebra, the derived and descending central series of a Lie algebra, nilpotent Lie algebra, solvable Lie algebra, solvable radical, semisimple and simple Lie algebra, maximal toral subalgebra, root system, irreducible root system.
- Give the definitions of and calculate with the classical Lie algebras.
- Describe the construction of the irreducible representations of sl_2.
- State Engel's Theorem, Lie's Theorem and Cartan's Criterion.
- Describe the direct sum decomposition into simple ideals and the Jordan-Chevalley decomposition for semisimple Lie algebras.
- Indicate how root systems correspond to semisimple Lie algebras and give the root space decomposition and root system of sl_n.