GROUPES ET ALGÈBRE DE LIE

The main objective of the course is to provide an introduction to Lie algebras, their representation theory, and their root space decomposition. This offers a glimpse at the classification theory of irreducible root systems. If time permits, many relations and generalizations could be discussed: relations to Lie groups, affine Lie algebras, integrable systems, and isomonodromy theory.

A preliminary outline of the chapters of the course is as follows: • Introduction : Lie algebras: definition, examples, and main properties. This chapter includes: structure constants, quotient algebras, centers, centralizers, normalizers, simple Lie algebras, and the adjoint representation.

• Solvable and nilpotent Lie algebras. This chapter includes: definitions and examples of solvable and nilpotent Lie algebras, Lie’s theorem and Engel’s theorem. • Semisimplicity: basic structures and representation theory. This chapter includes: solvability and semisimplicity, the Killing form, the basic structure of a semisimple Lie algebra, basic representation theory of Lie algebras. • Root space decomposition. This chapter includes: Cartan subalgebras, root space decomposition, uniqueness.
• Root systems. This chapter includes: abstract root systems, Cartan matrices and Dynkin diagrams. Weyl group actions (if time permits)

The literature is rather vast; a non-exhaustive list of references is given below [1] Roger Carter, Lie algebras of finite and affine type. CUP 2005. [2] Fulton Gonzalez, Lie Algebras, notes, 2007. [3] James Humphreys, Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 2012. [4] Anthony Knapp, Lie Groups Beyond an Introduction. Progress in Mathematics, Vol. 140. [5] Hans Samelson, Notes on Lie Algebras. Springer-Verlag, 1990. [6] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. Graduate Texts in Mathematics, 2013.