Representation Theory

1. Definitions and examples. Irreducible and completely reducible representations of finite groups.
2. Representations and modules. Simple and semisimple modules.
3. Applications to the group algebra. Maschke's Theorem.
4. Characters of finite groups: basic definitions and properties, irreducible characters, orthogonality relations, linear characters.
5. Character tables. Examples.
6. Applications of Character Theory. Solubility criteria, Burnside's Theorem, existence of normal subgroups and how to determine them.
7. Product of representations.
8. Induced representations and characters. Frobenius' Theorem.
9. Representations of symmetric groups. Partitions and Young tableaux, degrees of the irreducible representations of S_n.

Smooth manifolds, tangent space and tangent bundle of a smooth manifold, smooth vector fields and associated Lie algebra. Lie groups and associated Lie algebras: left invariant smooth vector fields. Functor between the category of Lie groups and that of Lie algebras.
Examples of Lie algebras. Adjoint representation. Ideals. Solvable, nilpotent and semisimple algebras. Theorems by Engel and Lie.
Killing form and characterization of semisimple algebras. Modules for Lie algebras and Weyl's Theorem on complete reducibility of modules for a semisimple Lie algebra. Modules of sl(2,C). Toral subalgebras and Cartan decomposition; roots systems.