Code: 02LIAG Lie Algebras and Lie Groups
Lecturer: doc. Ing. Libor Šnobl Ph.D. Weekly load: 3+2 Completion: A, EX
Department: 14102 Credits: 6 Semester: S
Definitions and properties of Lie groups and Lie algebras. Different types of Lie algebras, root systems and classification of complex simple Lie algebras. Introduction to theory of representations.
1. Lie group, Lie algebra and their relation.
2. Exponential map.
3. Subgroups and subalgebras, homogeneous spaces.
4. Universal covering.
5. Lie algebras - basic notions.
6. Killing form.
7. Theorems of Lie and Engel.
8. Cartan´s criteria.
9. Cartan´s subalgebra.
10. Root systems.
11. Classification of complex simple Lie algebras.
12. Representations of simple Lie algebras.
Seminar contents:
1. Groups GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(2n), Af(1).
2. Algebras gl(n), sl(n), o(n), so(n), u(n), su(n), sp(2n), af(1).
3. Connectedness and maximal torii of SU(n), SO(n).
4. Exponential map of sl(2) into SL(2).
5. Classification of Lie algebras up to dimension 3.
6. Killing form of Lie algebras up to dimension 3.
7. Root systems of A_l,B_l,C_l,D_l.
8. Tensor product of representations.
9. Representations of SU(3) and their relevance in particle physics.
Recommended literature:
Povinná literatura:
[1] D.H. Sattinger, O.L. Weaver: Lie Groups and Algebras, Springer Verlag 1986.
[2] A. P. Isaev, V. A. Rubakov: Theory Of Groups And Symmetries: Finite Groups, Lie Groups, And Lie Algebras, World Scientific 2018.

Doporučená literatura:
[3] H. Samelson: Notes on Lie algebras, Springer Verlag 1990.
[4] R. Gilmore: Lie groups, Physics and Geometry, CUP 2008.
[5] K. Erdmann, M.J. Wildon: Introduction to Lie Algebras, Springer Verlag 2006.
Lie groups, Lie algebras, classification of simple Lie algebras, representations of Lie algebras.

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