## Scope of questions for State exams of Master's degree program

Branch: Mathematical Physics

Subject: Lie Algebras, Lie Groups and Their Applications

Subjects regarding the questions:

- 02LIAG Lie Algebras and Lie Groups

1. Lie group and Lie algebra -definitions, exponential mapping, flow of levoinvariant vector field, matrix groups and algebras

2. Ambiguity in relation between Lie groups and algebras, classification of connected Lie groups with a given Lie algebra

3. Subgroups and subalgebras, actions of groups, cosets, isotropy subgroup, homogeneous spaces, examples of spaces and spacetimes with transitive actions as homogeneous spaces

4. Representation of Lie group / algebra, adjoint representation, irreducibility of representations, Schur lemma, examples of completely reducible representations

5. Basic classes of Lie algebras, Levi decomposition theorem into radical and semisimple Levi factor, classification of Lie algebras over R and C in dimensions 1, 2, 3 and their properties

6. Nilpotent Lie algebras, Engel theorem and its formulation for matrix Lie algebras

7. Solvable Lie algebras, Lie theorem, properties of derivable algebra

8. Killing's form, Cartan criteria of semisimplicity and solvability of the given algebra, decomposition of semisimple algebras into simple ideals

9. Cartan subalgebra and root system, their properties, Weyl-Chevalley normal form of semisimple Lie algebra, classification of simple Lie algebras over C, root and Dynkin diagrams

10. Finite-dimensional representations of simple Lie algebras over C, weights and weight diagrams, group SU(3) and its applications for elementary particle classification