Teachers: Francesco Esposito, Giovanna Carnovale
Schedule: tuesdays and wednesdays: 14.30-16.15, on fridays 11.30-13.15 starting on january 13th, 2008
Here is a schedule of the rooms where it will take place.

The course will be organized as a reading course, based on the book "Representations and invariants of the classical groups" by Goodman and Wallach. The aim of the course is to introduce the students to the basic concepts of algebraic groups up to the study of the rational finite dimensional representations through the structure theory, theory of Lie algebras and the highest weight theory. This will be treated by studying the model examples which are the so-called classical groups: the general linear groups, the special linear groups, the special orthogonal groups and the symplectic groups. The course is meant for students willing to read through the material and to lecture in English on the topics, guided by the teachers. Once a week the session will be devoted to the solutions of problems that are weekly assigned. For this reason active and regular participation is a necessary condition. The prerequisite is a good knowledge of linear algebra. Knowledge of differential manifolds, tangent spaces, vector fields and differential forms, and basic definitions of algebraic varieties is certainly welcome but not mandatory.

Here is an updated schedule of the lectures, the speakers and the chapters in the book that will be treated.

Reference texts:
R. Goodman and N.R. Wallach, "Representations and invariants of the classical groups"
J. Humphreys, Linear Algebraic groups
T. Springer, Linear Algebraic groups
Borel, Linear Algebraic groups

Zariski topology, connected and irreducible components, identity component of a linear algebraic group, algebric subgroups, affine algebraic groups are linear algebraic groups (1.1.4, 1.1.5, 1.1.6, V. Pastro)

Lie algebras; Lie algebra of a linear algebraic group; explicit description of the Lie algebras of the classical groups (1.2.1; 1.2.2, S. Virili)

Representations of a Lie algebra; the adjoint representation of a Lie algebra; differential of a representation and of a homomorphism and their properties; the adjoint representation of a linear algebraic group (1.2.3; 1.2.4, Carnovale)

Additive and multiplicative Jordan decomposition, Jordan-Chevalley decompositions and their behaviour with respect to representations (1.3.1, 1.3.3, Carnovale)

Regular morphisms from GL(1, C) to GL(n, C) (1.3.2, Carnovale) Maximal tori, conjugacy of maximal tori, the classical groups are generated by unipotent elements and they are connected (2.1.1, 2.1.2, A. Siviero)

Finite-dimensional irreducible representations of sl(2,C) and SL(2, C) (2.2.1, 2.2.1, J. Pellegrini)

Roots with respect to a maximal torus, commutation relations of root subspaces (2.3.1, 2.3.2, A. Fong)

Stucture of classical root systems, irreducibility of the adjoint representation for SO, Sp, SL (2.3.3, 2.3.4, C. Parolin)

Reductive groups, Casimir operator, Complete reducibility of the classical groups (algebraic proof) (2.4.1, 2.4.2, 2.4.3, N. Sambin)

The Weyl group for classical groups (2.5.1, 2.5.2, G.M. Dall'Ara)

The weight lattice, fundamental weights and dominant weights (2.5.3, 2.5.4, R. Padoan)

Highest weight theory: extreme vectors, highest weights, vectors annihilated by n, fundamental representations type An (5.1.1, 5.1.2, 5.1.3 F. Esposito)

Fundamental representations, Cartan product, weights of irreducible representations, A. Gnecchi)

Lowest weights and dual representations, Irreducible representations of GL(n, C) (5.1.6, 5.2.1, R. Rainone)

Compact real forms; the unitarian trick (1.4.2, 1.4.3, 2.4.4, O. Lenz)