Lie groups, Lie algebras, classical matrix groups GL(n,R), SO(n), SO(p,q), U(n), Lorentz group, Poincare group; exponential map, one-parameter subgroups; actions and basic representation theory, orbits and invariants; adjoint and coadjoint representations, Lie-Poisson bracket; solvable, nilpotent and semisimple Lie algebras
The students should be familiar with basic Linear Algebra, Calculus and Group Theory.
It is also assumed that they have some background in Elementary Topology including basic constructions related to manifolds (at least the definition).
1. Manifolds, vector fields, tangent bundle, smooth maps and diffeomorphisms
2. Lie groups and Lie algebras, relationship between them
3. Classical Lie groups GL(n,R), SL(n,R), O(n), O(p,q), U(n), SU(n), Sp(n,R) and their Lie algebras
4. Left and right invariant vector fields, one-parameter subgroups and exponential map
5. Fundamental group and universal covering of a Lie group
6. Actions of Lie groups
7. Homogeneous spaces and linear representations
8. Adjoint and co-adjoint representations
9. Solvable and nilpotent Lie groups, Lie and Engel theorems
10. Killing form, Cartan subalgebra, radical
11. Semisimple Lie algebras, classification (without detailed proof)
12. Basic facts on root systems and Dynkin diagrams