Entry requirements

120 credits including 90 credits Mathematic. Linear Algebra II. Modules and homological algebra taken. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• give an account of the following concepts: Lie algebra, representation of a Lie algebra, Cartan subalgebra, universal enveloping algebra, root system, Casimir element,
• define and exemplify the following Lie algebra concepts: nilpotent, solvable, and semisimple,
• formulate, know the general idea of the proof, and use the Poincaré-Birkhoff-Witt theorem, Engel's theorem, Lie's theorem and Weyl's theorem,
• provide examples of and do computations with representations of Lie algebras,
• explain how the representation theory of sl_2 motivates the notion of a root system and which role it plays in classification of semisimple Lie algebras.

Content

Lie algebras over the complex numbers, representations of Lie algebras, universal enveloping algebra and PBW Theorem, representations of sl_2, solvable, nilpotent and semi-simple Lie algebras, Engel's and Lie's theorems, Cartan subalgebras, root systems, classification of semisimple Lie algebras, Casimir element, Weyl's complete reducibility theorem.

Instruction

Lectures and problem solving sessions.

Assessment

Written assignments and oral test. 

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

The course 1MA332 Lie algebras cannot be included in the same degree as any of the courses 1MA048 and 1MA567.