**Quantum Groups and Infinite Dimensional Lie
Algebras**

**6510.001, Spring 2001**

**Conley**

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.