Math 559a Lie Groups and Lie Algebras

Instructor: John Palmer

Office: Math 718

Office Hours: Mon 11-12 Wed 2-3 Fri 2-3

email: palmer'at'math'dot'arizona'dot'edu

phone: 621 4364

Text: Representations of Finite and Compact Groups by Barry Simon

A Lie Group is a group that is also a manifold (with differentiable group operations). Calculus makes it possible to study some aspects of Lie Groups by studying the tangent space to the group at the identity. The group structure is reflected in the tangent space at the identity by a Lie Algebra structure on this vector space. Linear algebra is thus brought to bear on the study of Lie Groups.

A linear representation of a group, G, is a homomorphism of G into the group of invertible linear transformations of a vector space. The study of the different possible linear representations of a group is a subject that first got the attention of mathematicians in the late nineteenth century (mostly for the permutation group) and later the attention of physicists when it turned out that virtually all the symmetries in Quantum Physics are realized as linear representations of groups. In a remarkably prescient series of papers Hermann Weyl worked out the representation theory of the compact classical (Lie) groups just a year in advance of the quantum revolution in Physics. I propose (following Simon's book) to examine this representation theory in detail. Because Simon's book treats the representation theory of finite groups first this does mean that Lie Groups will not make an appearance until later in the term. I would like to briefly defend this choice. Groups like vector spaces are defined axiomatically. However, unlike vector spaces, groups have “personalities”. A finite dimensional vector space is determined (up to isomorphism) by a single number, its dimension. Groups, on the other hand, come in a staggering array of isomorphism classes. There are a number of ways to try to understand the “personality” of a group but representation theory has the virtue of bringing the highly developed subject of linear algebra to bear on subject (in addition to having many interesting applications in both mathematics and physics). Following Simon's treatment we will see the helpful parallels between the representation theory of finite and compact groups and eventually understand a reciprocity between the representation theory of the permutation group and the representation theory of the unitary group that plays out in the different symmetry classes of tensors. The Frobenius character formula for the representations of the permutation group (mysterious) is deduced from the Weyl character formula for the unitary group (much less mysterious).

It is possible (and even attractive) to develop the general theory of the relationship between Lie Groups and their Lie Algebras from the abstract manifold point of view. This is a wonderful exercise in geometry and analysis but suffers a bit from being so general that the individual personalities of the different Lie Groups are not much in evidence (until, of course, one starts asking sufficiently detailed questions). We will see some of this relationship but it will mostly be in the context of our concentration on the “classical groups”, like the unitary group, the symplectic group, and etc.

If this course continues next term we may have time to look at some applications of representation theory to gauge theories in physics. This is another reason why I thought that concentrating on representation theory would be entertaining. The modern theories of the electromagnetic, weak and strong interactions are all gauge theories in which representation theory is a central player.

The course will have regularly assigned homework counting 30% of your final grade, a mid term exam counting 30% and a final exam counting 40%. I will announce reading assignments in class and I will distribute homework sets on this site.

Problem set #1: problem set 1

Problem set #2 : problem set 2

Problem set #3 : problem set 3

Problem set #4 : problem set 4

Problem set #5 : Problem set #5

Remarks on Homework 4 Remarks on Homework 4

Problem set #6 : problem set 6

Takehome final: final exam

Problem set #7: Problem set 7

Problem set #8: Problem set 8

Final Exam