||8 credit points
||Opdam, E.M. (Universiteit van Amsterdam), Stokman, J.V. (Universiteit van Amsterdam)
||To obtain a firm basic knowledge on Lie algebras, the classification of complex semisimple Lie algebras, and the associated representation theory.
||The course provides an introduction to the theory of semisimple Lie algebras. Lie algebras are linear approximations of infinite smooth groups. Semisimple Lie algebras form an important subclass which play a profound role in problems of mathematics and physics in the presence of symmetries.
We start with the concept of a Lie algebra and provide many classes of examples. We proceed by studying nilpotent and solvable Lie algebras. Subsequently we focus our attention to complex semisimple Lie algebras. We discuss the associated structure theory, leading to their classification in terms of Dynkin diagrams.
We establish the fundamental results on the representation
theory of complex semisimple Lie algebras. This includes a discussion of universal enveloping algebras and Weyl's character formula. We also discuss the algebraic group of adjoint type associated to a complex semisimple Lie algebra.
||There are weekly lectures (3 x 45 minutes).
A small part of the lectures will be used to discuss (homework) exercises.
||There will be a take home exam which will contribute 80% to the final mark. During the course homework exercises have to be handed in, which will contribute 20% to the final mark.
||J.E. Humphreys, " Introduction to Lie algebras and representation
Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. xii+171 pp.
Possibly additional material will be made available during the course in the form of downloadable pdf documents.
||bachelor mathematics or physics. Good knowledge of basic algebra and linear algebra.