Complex semi-simple Lie algebras and their representations

under construction

In this course we learn the basics of w: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

some references

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.

• google: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

tests

• ...

examination

• ...

beyond

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