*Note that this course is cross-listed with the ELTE University and is held on their campus. Its schedule is set by ELTE.*

*Instructor:* Dr. Péter Pál Pálfy

*Text:*J.E. Humphreys, Introduction to Lie Algebras and Representation Theory,
Graduate Texts in Mathematics, vol. 9, Springer, 1972

*Prerequisites:*
Some familiarity with standard linear algebra and basic
algebraic structures (groups, rings, fields) is expected from the
students.

*Course description:*
Lie algebras belong to the most important types of structures in
mathematics. In this course they will be studied from the algebraic point
of view. This course can serve as an introduction to a course on Lie
groups or on finite simple groups of Lie type.
*Topics:*

- Definition and basic properties of Lie algebras.
- Derivations, Killing form.
- Classical Lie algebras.
- Nilpotent and solvable Lie algebras. Theorems of Engel and Lie.
- Cartan criterion. Cartan subalgebra.
- Semisimiple Lie algebras, roots, root systems, Weyl group, Cartan matrix, Dynkin diagram.
- The simple Lie algebras.
- Chevalley basis.
- Enveloping algebras. Poincaré-Birkhoff-Witt theorem.
- Free Lie algebras, Witt formula. Baker-Campbell-Hausdorff formula.
- Representations of Lie algebras. Casimir element. Weyl's theorem.
- Representations of sl(2,C).