Lie groups
Prof. Paul Gunnells, LGRT 1115L, 413.545.6009, gunnells at math dot umass
dot edu. Email to this address is the best way to contact me.
Mondays and Wednesdays,
8:00-9:00, and by appointment.
A Lie group is a smooth manifold that is a group and such that the
group operations (multiplication and inversion) are smooth maps.
Their study involves a pleasant mix of geometry, algebra, analysis,
and combinatorics.
The goals of this course are to give an overview of some of the most
important aspects of Lie groups, including structure theory and
representation theory. I hope that at the end of the course you will
have a feel for the mechanics of Lie groups and Lie algebras that show
up in "real life," and (more concretely) will know what the characters
of representations are and how to compute them.
The course will be divided into three parts.
- The first part will cover general material, including basic
definitions and examples, real and complex Lie groups, compact Lie
groups, Lie algebras, the classical groups, and the exponential map.
- The second part will focus on the structure theory of semisimple complex
Lie algebras, including Cartan subalgebras, root systems, Dynkin
diagrams, and the classification theorem.
- The final part will discuss representation theory, including
weights, characters, and character formulas. I hope to discuss some
combinatorial models for representations (path model/crystal
graphs).
Hall, Brian C. Lie groups, Lie algebras, and representations. An
elementary introduction. Graduate Texts in Mathematics
222. Springer-Verlag, New York, 2003. The text is not required
(exercises will be assigned separately), but the course will roughly
follow its presentation.
- Humphreys, James E. Introduction to Lie algebras and representation
theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New
York-Berlin, 1978. A classic. Would have been my choice for a
textbook, but unfortunately only covers Lie algebras.
- Fulton, William; Harris, Joe. Representation theory. A first
course. Graduate Texts in Mathematics, 129. Readings in
Mathematics. Springer-Verlag, New York, 1991. A beautiful book to
read. Very useful for self-study.
- Bump, Daniel. Lie groups. Graduate Texts in Mathematics,
225. Springer-Verlag, New York, 2004. Perhaps too hard for beginners,
but it contains an excellent collection of topics in the final part.
- Varadarajan, V. S. Lie groups, Lie algebras, and their
representations. Graduate Texts in Mathematics,
102. Springer-Verlag, New York, 1984. Another classic. Very
comprehensive. Much groups. Such wow.
- Representation theory of Lie groups. Proceedings of the SRC/LMS
Research Symposium held in Oxford, June 28--July 15, 1977. Edited by
G. L. Luke. London Mathematical Society Lecture Note Series,
34. Cambridge University Press, Cambridge-New York, 1979. See
especially the articles by Macdonald and Bott.
- Onishchik, A. L.; Vinberg, E. B. Lie groups and algebraic groups.
Translated by D. A. Leites. Springer Series in Soviet
Mathematics. Springer-Verlag, Berlin, 1990. Written with a more
algebraic flavor. Takes the unusual approach of omitting almost all
proofs and presenting the material as a series of exercies. (This
is not as crazy as it sounds. In fact it's a very pleasant read.)
- Knapp, Anthony W. Lie groups beyond an introduction. Second
edition. Progress in Mathematics, 140. Birkhauser Boston, Inc.,
Boston, MA, 2002. Contains a lot of material with complete proofs.
Thorough, but difficult to read if this is your first exposure.
- Springer, T. A. Linear algebraic groups. Second edition. Progress in
Mathematics, 9. Birkhauser Boston, Inc., Boston, MA, 1998. Sure,
it's a textbook on algebraic groups, but there's plenty of
relevance for the study of Lie groups.
- Freudenthal, Hans; de Vries, H. Linear Lie groups. Pure and Applied
Mathematics, Vol. 35 Academic Press, New York-London 1969. Bizarre and fascinating.
Lie groups are a central topic; there are many freely available sets
of lecture notes floating around. Here are a few I found. I haven't
read them, so can't give comments.
- The program LiE can
compute tons of data for Lie groups and Lie algebras, including just
about anything for root systems, characters of representations, etc.
It's easy to install on a Linux system; not sure about others. The user manual actually
gives a pretty good overview of representation theory.
- The system Sage includes a lot of the
same functionality, along with tons of other stuff. Should be easy
to install on any modern computer.
Here are
some helpful pages to get you going with material relevant for our course.
- The Atlas of Lie
groups and representations. This is mainly focussed on
infinite-dimensional representations, but there is a lot of material
here relevant to our course (check out the root systems applet).
The grades for this course will be based equally on exercises and
class participation. As in a previous graduate course, I won't assign
individual problem sets, but instead will maintain a
list of problems. You can do them at any time during the course; the list
will be updated throughout the term (including bugfixes). I would
like you to complete 20 problems. Let me know if you find any typos
in the problems, or if something isn't clear.
Revised: Wed Apr 23 10:25:11 EDT 2014
Paul Gunnells
gunnells at math dot umass dot edu