# 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? edit

- 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 edit

### on paper edit

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 edit

There is plenty of lecture notes and other good references on line to suit every taste.

## lessons edit

- 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 edit

- ...

## examination edit

- ...