Notes for 128: Combinatorial Representation Theory of Complex Lie Algebras and Related Topics

Recommended reading 2 1. The poster child of CRT: the symmetric group 2 1.1. Our best chance of understanding big bad algebraic structures: representations! 3 1.2. Where is this all going? 6 2. Lie algebras 7 2.1. Favorite Examples 7 2.2. Categories, Functors, and the Universal Enveloping Algebra 9 3. Representations of g: a first try 10 3.1. Dual spaces and Hopf algebras: lessons from Group theory 10 3.2. Representations of sl2(C) 12 4. Finite dimensional complex semisimple Lie algebras 15 4.1. Forms: symmetric, bilinear, invariant, and non-degenerate 16 4.2. Jordan-Chevalley decomposition, and lots of sl2’s 17 4.3. Cartan subalgebras and roots 18 5. Highest weight representations 27 6. More on roots and bases 33 6.1. Abstract root systems, Coxeter diagrams, and Dynkin diagrams 36 7. Weyl groups 40 8. Back to representation theory 44 8.1. The Universal Casimir element and Freudenthal’s multiplicity formula 46 8.2. Weyl character formula 48 8.3. Path model 52 9. Centralizer algebras 62 9.1. First example: Type Ar and the symmetric group 64 9.2. Back to idempotents 67 Appendix A. Bases, roots, and weights for the classical Lie algebras 69 A.1. Type Ar 69 A.2. Type Cr 70 Index 72 References 74