Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 846: Introduction to quantum groups


 Spring 2019

Title: Introduction to quantum groups

Course No. 846

Time: MWF 11:00 MWF

Instructor:  Paul Terwilliger

Prerequisite: Good understanding of linear algebra

Textbook: Lectures on quantum groups, by Jens Carsten Jantzen, Graduate
Studies in Mathematics Vol. 6
ISBN-13: 978-0821804780 ISBN-10: 0821804782

DESCRIPTION: In this introductory course
we will discuss the basic concepts associated
with quantum groups.

We will begin with a concrete example: the quantum
group U_q(sl_2). We will define this algebra via
generators and relations; we will obtain a basis; we
will compute the center, and we will describe the
finite dimensional modules. We will discuss how U_q(sl_2)
is a quantized enveloping algebra for the Lie algebra
sl_2. We will discuss how U_q(sl_2) has the structure
of a Hopf algebra.

With the example of U_q(sl_2) in mind, we will turn
our attention to the quantum group U_q(g), where g
is a finite dimensional complex semisimple Lie algebra.
We will develop the theory of U_q(g) from first principles.
Along the way we will encounter the following topics:
The quantum trace; the Yang-Baxter equation;
the triangular decomposition of U_q(g); modules for U_q(g);
the center of U_q(g); the Harish-Chandra homomorphism;
the Hopf algebra structure for U_q(g); R-matrices;
a bilinear form which pairs the positive and negative
parts of U_q(g); the braid group action and PBW type basis;
crystal bases.

This course is recommended for anyone interested in
Lie theory, algebraic combinatorics, special functions,
knot invariants, and statistical mechanical models.

A good understanding of undergraduate linear algebra.

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, WI  53706

(608) 263-3054

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