Representation Theory

Prof. Cesare Reina

  1. Finite groups: Definitions. Examples: symmertic and alternating groups, cyclic and diedral groups, Platonic groups, the group of the Rubik
  2. sl(2,C): Roots and weights. Finite dimensional irreducible representations. The projective line. Projective spaces and rational normal curves.
  3. sl(3,C): Roots and weigths. Finite dimensional irreducible reprentations. The projective plane. The flag variety F(1,2;3).
  4. Representations af semisimple Lie algebras: Complex Lie algebras. Finite dimensional irreducible representations. Dynkin diagrams and classification.
  5. 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.
  6. Quantum groups: Algebraic groups anf Hopf algebras. The quantum group Sl_q(n). Sl_q(2) at roots of unity.