MTH424: Introduction to Lie algebras

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Lie algebras: definition and examples, ideals and homomorphisms, quotient algebras, low dimensional Lie algebras.
Universal enveloping algebras, Poincare-Birkhoff-Witt theorem.
Solvable and nilpotent Lie algebras, Engels theorem, Lies theorem.
Representations of Lie algebras, Schurs lemma, representations of sl(2,C).
Killing form, Cartans criteria for solvability and semisimplicity, derivations of semisimple Lie algebras.
Cartan subalgebras, root space decomposition, Cartan subalgebras as inner product spaces.
Root systems, Weyl group of a root system, Dynkin diagrams.
Classical Lie algebras sl(n,C),so(n,C),sp(n,C).
Classification of root systems, irreducible root systems and complex simple Lie algebras.