Complex semi-simple Lie algebras and their representations

under construction

In this course we learn the basics of complex semi-simple Lie algebras.

Why?

 * the classification of compact Lie groups reduces to the classification of semi-simple Lie algebras
 * the key notion of a root system reappears in many branches of mathematics and theoretical physics
 * Lie algebras and their representations are intimately related to quantum mechanics

on paper
In increasing order of details:
 * J.-P. Serre, Complex semi-simple Lie algebras (translated from French: Algebres de Lie complex semi-simple
 * J. E. Humphreys, Introduction to Lie algebras and representation theory, ISBN 978-0-387-90053-7
 * W. Fulton, J. Harris, Representation theory, A first course

on line
There is plenty of lecture notes and other good references on line to suit every taste.
 * semisimple+Lie+algebras

lessons

 * Lie algebra
 * Linear Lie algebras
 * /derivations and automorphisms
 * soluable and nilpotent Lie algebras
 * representations of Lie algebras
 * examples: classical Lie algebras
 * example: representations of sl2
 * Lie's theorem and Engel's theorem
 * Schur's lemma
 * Casimir element
 * Weyl's theorem on complex reducibility
 * Killing form
 * Cartan subalgebra
 * Borel subalgebra
 * roots and weights of a Lie algebra
 * root system
 * Cartan matrix
 * Dynkin diagram
 * Verma module
 * Harish-Chandra's theorem
 * Weyl character formula
 * Kostant's multiplicity formula
 * Poincare-Birkhoff-Witt theorem

beyond

 * Lie group
 * algebraic group
 * Kac-Moody algebra
 * quantum algebra
 * vertex algebra