Quantum Groups and Infinite Dimensional Lie Algebras
6510.001,  Spring 2001

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the University of North Texas.

INSTRUCTOR:   Charles  Conley,  GAB  475,  (940)  565 - 3326

OFFICE HOURS:   Monday 2:00-3:00, Wednesday 10:00-11:00 and 2:00-3:00, Friday 12:00-1:00

CLASS MEETS:   Monday and Wednesday, 12:30 - 1:50, GAB 438

PREREQUISITES:   My course on Lie algebras (6510, Fall 2000) is not a prerequisite, and no prior knowledge of Lie algebras, their representations, or infinite dimensional vector spaces will be assumed.  However, you will need a solid background in finite dimensional linear algebra.  If you would like to try the class but are not sure you are prepared, please discuss it with me.

REFERENCES:   There is no introductory book covering all of our subjects.  Significant parts of the course will be based on ``Infinite Dimensional Lie Algebras,'' by V. Kac, and ``Lectures on Quantum Groups,'' by J. C. Jantzen, and I may also use the books on quantum groups by Kassel, Chari and Pressley, and H. H. Andersen.  These are all quite difficult; the notes will go more slowly and fill in more details.  You can find other texts by searching the UNT library for the two key phrases in the course title, and if you find a topic you would like to learn, let me know and I will try to include it.

PLAN:   The course will be an introduction to several interrelated subjects of current research interest: quantum groups, Kac-Moody Lie algebras, the Virasoro Lie algebra, Lie super algebras, and Lie algebras in characteristic p.  These are advanced topics, and so we will learn them by considering the simplest non-trivial example in each case, namely, the appropriate generalization of the 3-dimensional Lie algebra sl(2).  The emphasis will always be on the representations of the algebra in question, and so we will begin with the representation theory of sl(2) itself.  Then we will study:

1.   The smallest simple Lie superalgebra, osp(1|2), a 5-dimensional algebra containing sl(2).  The two new dimensions form a ``square root'' of sl(2), which turns out to link its even and odd dimensional irreducible representations.  Lie superalgebras arose from relations between bosons and fermions in physics.

2.   The Virasoro Lie algebra, an extension of the Lie algebra of polynomial vector fields on the circle, which has basis {z^n(d/dz): n in Z}.  It is of interest because it acts by derivations on the Kac-Moody Lie algebras.  It contains an important subalgebra isomorphic to sl(2): the infinitesimal linear fractional transformations, which are spanned by z^n(d/dz) for n = 0, 1, 2.

3.   The affine Kac-Moody Lie algebra ^L(sl(2)), an extension of the Lie algebra of polynomial maps from the circle to sl(2).  The Kac-Moody Lie algebras have become very important in many areas of mathematics and physics over the past three decades.

4.   The quantum group U_q(sl(2)), which is not a group, but an algebra obtained by deforming the universal enveloping algebra of sl(2).  It is in fact a Hopf algebra; we will learn what this means, and acquaint ourselves with the Yang-Baxter equations.  There is an important dichotomy in the representation theory of U_q(sl(2)),
between q a root of unity and q not a root of unity, and we will consider both cases.  Quantum groups were introduced about twenty years ago to explain new types of symmetries in physics.

5.   The finite Lie algebra sl(2, Z_p) of characteristic p, whose representations in characteristic p are related to the representations of U_q(sl(2)) for q^p = 1.

As we progress, we will need to develop some theoretical tools, for example, tensor products of vector spaces, universal enveloping algebras of Lie algebras, the Poincare-Birkhoff-Witt theorem, and Verma modules.  We will do this as concretely as our examples allow.  If there is time, we will briefly discuss the representation theory of sl(2) over p-adic fields, and perhaps also deformation quantization.

It is the responsibility of students with certified disabilities to provide the instructor with appropriate documentation from the Dean of Students Office.