Lie Algebras - M1 - 8EC

1) Prerequisites

  • Group and ring theory as covered in basic algebra courses in the Bachelor Mathematics (Sections I.1-5,7; II.1-3; III.1-6; IV.1-3; V.1,2,5 in Lang, “Algebra”, Graduate Texts in Math. 221, Springer).

  • Tensor products of modules over rings (Chapter XVI in Lang, “Algebra”, Graduate Texts in Math. 221, Springer).

  • Knowledge of Lie groups and representation theory of finite groups is not necessary, but is helpful. 

2) Aim of the course

A Lie algebra is a vector space endowed with a binary operation generalising the commutator bracket for associative algebras. Semisimple Lie algebras form an important subclass of Lie algebras that abundantly appear in mathematical and physical contexts involving symmetries. This course gives an introduction to the theory of semisimple Lie algebras and their representation theory. Topics covered in the course are: nilpotent and solvable Lie algebras, structure theory and classification of semisimple Lie algebras, universal enveloping algebras and representation theory of semisimple Lie algebras. 

This course entirely focusses on the algebraic aspects of Lie algebras. The analytic aspects, which arise from the fact that real and complex Lie algebras provide infinitesimal descriptions of smooth manifolds with compatible group structure, are part of the mastermath course Lie Groups.

3) Rules about Homework/Exam

There will be biweekly homework exercises and final written exam. The biweekly homework counts for 30% of the final grade unless the mark of the final written exam is higher than the average mark of the homework exercises, in which case the homework is discarded. The homework will also be discarded when the mark of the final written exam is less than 5.0 and in case of a re-exam. The re-exam will be a written exam.

 

4) Lecture notes/Literature

We will follow the book

J.E. Humphreys, " Introduction to Lie algebras and representation theory", Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. xii+171 pp.

Additional recommended reading material is

A. Kirillov, Jr., "An Introduction to Lie groups and Lie algebras", Cambridge studies in advanced mathematics 113.

 

5) Lecturers

Eric Opdam and Jasper Stokman