
MAT 552
Lie groups, Lie algebras and their representations

Place and time
MF 09:25  10:45, Mon Math 4130, Fri SBS N102
Instructor
Sorin Popescu
(office: Math 4119, tel. 6328358, email
sorin@math.sunysb.edu)
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 III), MAT 534, MAT 535 (Algebra
III), as well as certain basic facts from
MAT 544 (Analysis). It should however be possible to fill in some of the gaps during
the semester.
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 semisimple
representations. Schur lemma
 Compact groups. Haar measure; complete reducibility of representations.
Representations of S^{1}.
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: sl_{2} 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 sl_{2}, SO(3) and spectrum of the hydrogen atom
 Highest weight representations. Verma modules
 Weyl character formula
 Example: representations of sl_{n}
I will assign problems in each lecture, ranging in difficulty from
routine to more challenging. There will be also a takehome midterm and
a final exam. Course grades will be based on these exams and homework
problems (and any other participation).
Sorin Popescu
20020825