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).