Introduction to Lie algebras

Spring 2014, IMPA

Instructor: Reimundo Heluani

Office: 432

Homework

Lecture 1 19/8
Basic definitions: algebras, Lie algebras. Examples: gl(V), sl(V), classical Lie algebras. Subalgebras, ideals, morphisms, Kernels, quotients.

Lecture 2 21/8
Algebraic Groups and their Lie algebras. Inner derivations. Representations of Lie algebras. Center of a Lie algebra. Low dimensional examples.

Lecture 3 26/8
Nilpotent operators. Lower central series and derived series. Nilpotent and solvable Lie algebras. Engel's Theorem.

Lecture 4 28/8
Engel's criterion. Weight spaces. Lie's lemma and theorem. Invariant flags. Generalized weight spaces.

Lecture 5 02/9
Recall Jordan decomposition. Generalized weight spaces. Weight spaces for Nilpotent Lie algebras.

Lecture 6 04/9
Characteristic polynomial: rank and regular elements. Zariski topology. Discriminants. Normalizers. Cartan subalgebras.

Lecture 7 09/9
Generalized weight space decompositions with respect to a Cartan subalgebra. Existence and dimensions of Cartan subalgebras. Inner derivations and automorphisms.

Lecture 8 11/9
Trace forms. Cartan's criterion. Semi-simple Lie algebras. Radicals.

Lecture 9 16/9
Linear semisimple Lie algebras. Non-degenerate forms. Decomposition into simple algebras.