## Representation theory: an introduction

### Prof. Cesare Reina

Content:

- Basic facts from algebra: groups, homomorphisms, exact sequences; rings, modules, algebras; linear representations; Schur's lemma; multilinear algebra.
- 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)).
- SL(2,C) and the projective line: basic facts about SL(2,C); the complex projective line; irreducible representations and rational normal curves.
- 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).
- 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.
- Representation theory and homogeneous spaces: compact forms; group actions and homogeneous spaces; homogeneous vector bundles and induced representations; the Borel-Weil-Bott theorem.
- Affine group schemes: algebraic groups; affine group schemes and Hopf algebras; comodules and the left regular representation; homogeneous schemes.
- 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)