1. komponenta

Lecture type	Total
Lectures	45

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The theory of Lie algebras is important in the group theory, mathematical physics, geometry and other mathematical theories. The aim of this course to introduce students with elements of the theory of Lie algebras and to present some methods and results in the representation theory of simple Lie algebras.

COURSE DESCRIPTION AND SYLLABUS:
1. Definition and examples of Lie algebras
2. Ideals, subalgebras and homomorphisms. Representations of Lie algebras.
3. Solvable and nilpotent Lie algebras. Engel's Theorem
4. Semisimple Lie algebras. Theorems of Lie and Cartan. Killing form.
5. Complete reducibility of representations. Weyl's theorem.
6. Representations of the Lie algebra sl(2,F)
7. Root space decomposition of a semisimple Lie algebra. Examples.
8. Root systems. Dynkin diagram.
9. Isomorphism and conjugacy theorems.
10. Universal enveloping algebra.
11. Finite dimensional representations of simple Lie algebras.
12. Highest weight representations. Verma modules.
13. Character formulas

1. Introduction to Lie algebras and representation theory, 1. J. E. Humphreys, Springer Verlag, 1978.
2. Lie groups and Lie algebras, Chapters 1 - 3, N. Bourbaki, Springer Verlag, 1998.
3. Lie groups and Lie algebras, Chapters 4 - 6, N. Bourbaki, Springer Verlag, 2002.
4. Lectures on Lie groups and Lie algebras, R. Carter, G. Segal, I. Macdonald,, Cambridge University Press, 1995.
5. Lie Algebras, N. Jacobson, Dover, New York, 1979.

Izborni predmet 1, 2 - Regular study - Theoretical Mathematics

Izborni predmet 1, 2 - Regular study - Theoretical Mathematics

Izborni predmet 3, 4 - Regular study - Theoretical Mathematics

Izborni predmet 3, 4 - Regular study - Theoretical Mathematics