## Representation Theory

### Prof. Cesare Reina

- Finite groups: Definitions. Examples: symmertic and alternating
groups, cyclic and diedral groups, Platonic groups, the group of the Rubik
- sl(2,C): Roots and weights. Finite dimensional irreducible
representations. The projective line. Projective spaces and rational
normal curves.
- sl(3,C): Roots and weigths. Finite dimensional irreducible
reprentations. The projective plane. The flag variety F(1,2;3).
- Representations af semisimple Lie algebras: Complex Lie algebras.
Finite dimensional irreducible representations. Dynkin diagrams and
classification.
- Representation theory and homogeneous spaces: Lie algebras and
Lie groups. Compact forms. Homogeneous spaces. Parabolic subgroups and
compact complex homogeneous spaces. Homogeneous vector bundles and induced
representations. Borel-Weil-Bott theorem.
- Quantum groups: Algebraic groups anf Hopf algebras. The quantum
group Sl_q(n). Sl_q(2) at roots of unity.

### References:

- I.P Serre "Linear Representations of finite groups" Springer-Verlag (1977
- W. Fulton, J Harris "Representation Theory - a First Course" Springer-Verlag (1991)