Lie groups, Lie algebras and their representations
Place and time
MF 09:25 - 10:45, Mon Math 4-130, Fri SBS N-102
(office: Math 4-119, tel. 632-8358, e-mail
Course starts on September 9!
This course will cover basic theory of Lie groups and Lie algebras.
One of its aims is to provide a brief introduction to modern representation theory
by working with the classical examples: the general linear, orthogonal and
symplectic Lie groups.
The framework will be mainly algebraic, and/or sometimes analytic.
We will assume material covered in MAT 530, MAT 531
(Geometry/Topology I-II), MAT 534, MAT 535 (Algebra
I-II), as well as certain basic facts from
MAT 544 (Analysis). It should however be possible to fill in some of the gaps during
The basic textbooks are:
Both books will be on reserve in Math/Phys library.
In addition, you may find the following books useful:
- J. F. Adams, Lectures on Lie Groups - a "classic"
- J.-P. Serre, Lie algebras and Lie groups - an excellent exposition
of the theory of simple Lie algebras
- Brocker and tom Dieck, Representation of compact Lie groups.
- A. Knapp, Lie groups beyond an introduction, Birkhauser, 1996
- Chevalley, Theory of Lie Groups I.
The following is a tentative list of what we will try to cover
in class. Actual material will depend also on the interests of the course participants.
Part I. Lie groups
- Basic definitions; examples.
- Linear groups; exponential
mapping. Classical groups: SL, SO, Sp, SU.
- Group representations; adjoint representation. Simple and semi-simple
representations. Schur lemma
- Compact groups. Haar measure; complete reducibility of representations.
Representations of S1.
Part II. Lie algebras: basic definitions
- Lie algebra of a Lie group: different definitions.
- Lie theory; exponential mapping, third Lie theorem
- Solvable, simple and semisimple Lie algebras
- Killing form; criterion of semisimplicity. Real forms and complexification.
Part III. Structure theory of simple Lie algebras
- Example: sl2 and its representations
- Cartan subalgebra, roots
- Abstract root systems, Cartan matrices, Weyl group. Dynkin diagrams
- Correspondence between root systems and simple Lie algebras
Part IV. Representations of simple Lie algebras.
- Representations of sl2, SO(3) and spectrum of the hydrogen atom
- Highest weight representations. Verma modules
- Weyl character formula
- Example: representations of sln
I will assign problems in each lecture, ranging in difficulty from
routine to more challenging. There will be also a take-home midterm and
a final exam. Course grades will be based on these exams and homework
problems (and any other participation).