Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Algebra and Number Theory

Code

liealg1u0um17em

Title

Lie algebras (l)

Usual semester

Spring

Published semester

2025/26/2

ECTS

3

Language

en

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyse mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

We examine one of the most important structures, Lie algebras from an algebraic standpoint. This course may serve as an introduction to the study of Lie groups or finite simple groups of Lie type. Definition and basic properties of Lie algebras. Derivations, Killing form.Classical Lie algebras. Nilpotent and solvable Lie algebras.Theorems of Engel and Lie. Cartan’s criterion. Cartan subalgebra. Semisimple Lie algebras, roots, root systems, Weyl group, Cartan matrix, Dynkin diagram. Simple Lie algebras, Chevalley basis. Enveloping algebra, the Poincaré–Birkhoff–Witt theorem. Free Lie algebras, Witt’s formula. The Baker–Campbell–Hausdorff formula. Representations, Casimir element, Weyl’s theorem, representations of sl(2,C).

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

Humphreys, J.E.: Introduction to Lie algebras and representation theory. Springer

Programmes of the course