Introduction to Lie groups and Lie algebras

1. Review of differential manifolds
Basic notions of differential geometry. Vector fields and differential forms. Lie derivative. Cartan calculus. Stokes’
theorem. Fiber bundles.
2. Lie groups
General definitions and basics properties. Classical Lie groups. Lie subgroup. Lie morphism. Lie algebra associated
to a Lie group. Left-invariant vector fields and forms. Maurer-Cartan equations. Exponential map.
3. Lie algebras
Basic definitions. Ideals and Lie subalgebras. Lie theorems. Real and complex forms. Universal enveloping
algebra. Poincaré-Birkhoff-Witt theorem. Campbell-Hausdorff formula.
4. Structure of Lie algebras
Solvable, nilpotent and semisimple Lie algebras. Killing form. Lie and Engel theorems. Cartan criterion. Jordan
decomposition.
5. Semisimple Lie algebras
Cartan subalgebra. Root system. Dynkin diagram. Classification of simple Lie algebras. Weyl formula. Finite dimensional
representations of sl(2).