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.
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
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