## Representation Theory: an introduction

### Cesare Reina

0) BASIC FACTS FROM ALGEBRA: Groups, homomorphisms, exact sequences. Rings, modules, algebras. Linear representations. Schur's lemma. Multilinear algebra.

1) FINITE GROUPS: Cosets. Lagrange's theorem. Simple groups. Extensions. Transformation groups. Examples: cyclic groups, symmetric groups, alternating group, the group of Rubik's cube, finite subgroups of SO(2) and SO(3)

2) SL(2,C) AND THE PROJECTIVE LINE: Basic facts about SL(2,C). The complex projective line. Irreducible representations and rational normal curves.

3) SL(3,C), THE PROJECTIVE PLANE AND THE FLAG MANIFOLD: Basic facts about SL(3,C). The projective plane and its dual. The flag manifold F(1,2:3)

4) REPRESENTATIONS OF SEMI-SIMPLE LIE ALGEBRAS: Basic facts about Lie algebras. Cartan decomposition. Finite dimensional irreducible representations. Dynkin diagrams and the classification of simple Lie algebras.

5) REPRESENTATION THEORY AND HOMOGENEOUS SPACES: Compact forms. Group actions and homogeneous spaces. Homogeneous vector bundles and induced representations. The Borel-Weil-Bott theorem.

6) AFFINE GROUP SCHEMES: Alebraic groups. Affine group schemes and Hopf algebras. Comodules and the left regular representation. Homogeneous schemes.

7) QUANTUM GROUPS: The quantum group SL_q(n). SL_q(2) at roots of unity and the quantum projective line.

References:
- A. Borel, Linear algebraic groups GTM 126 Springer (1991)
- N. Bourbaki, Gropes del Lie at algebres de Lie, Ch.s I-VII, Hermann (1960-75)
- M. L. Curtis, Matrix Groups, Springer
- W. Fulton, J. Harris, Representation theory, GTM, Springer (1991)
- J. P. Serre, Linear representations of finite groups, GTM 42, Springer (1977)