Topics in Ring and Representation Theory 2015/16 Schedule.
The following is the schedule of the lectures. It will be regularly updated as the course progresses.
1. Definition of a Lie algebra, background, subalgebras and examples.
2. Ideals, homomorphisms and first properties. Why Lie algebras? (up to end of Chapter 1)
3. Constructions with ideals, quotients and isomorphism theorems.
4. Correspondence Theorem for ideals, direct sums and examples. (up to end of Chapter 2). Short summary of Chapter 3 (ignoring most of it)
5. Solvable Lie algebras and their properties.
6. Radicals, semisimple, nilpotent Lie algebras. Structure of lectures ahead. (up to end of Chapter 4)
7. Nilpotent maps and weights.
8. The Invariance Lemma (up to end of Chapter 5) and Engel Part 1.
9. Engel's Theorem 1 and 2 and proof, and Lie's Theorem
10. End of proof of Lie's Theorem (up to end of Chapter 6), then representations and examples, Modules, homomorphisms, factors etc, and why there are the same as representations.
11. Irreducible, indecomposable modules, Schur's lemma (finishing Chapter 7). Summary of representation theory of sl2 (most of this will be covered at Workshops).
12. Cartan's 1st criterion. The Killing form and its properties.
13. Cartan's 2nd criterion and first applications (end of Chapter 9).
14. sl2 and sl3: how to draw a hexagon!
15. Root Space Decompositions and Cartan subalgebras. First properties of roots.
16. Enter sl2. Subalgebras and root strings, and proof root spaces are one-dimensional.
17. Finish proof root spaces are one-dimensional. Other properties of roots and their number.
18. Towards Dynkin diagrams. Real inner product spaces from semisimple Lie algebras, and summary of root systems.
19. Putting it all together: semisimple Lie algebras are classified by Dynkin diagrams.
20. Revision lecture.