Representation theory: an introduction

Prof. Cesare Reina


  1. Basic facts from algebra: groups, homomorphisms, exact sequences; rings, modules, algebras; linear representations; Schur's lemma; multilinear algebra.
  2. 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)).
  3. SL(2,C) and the projective line: basic facts about SL(2,C); the complex projective line; irreducible representations and rational normal curves.
  4. 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).
  5. 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.
  6. Representation theory and homogeneous spaces: compact forms; group actions and homogeneous spaces; homogeneous vector bundles and induced representations; the Borel-Weil-Bott theorem.
  7. Affine group schemes: algebraic groups; affine group schemes and Hopf algebras; comodules and the left regular representation; homogeneous schemes.
  8. Quantum groups: the quantum group SLq(n); SLq(2) at roots of unity and the quantum projective line.