Fall 2015
Course No. 846
Course title: Quantum groups and Hopf algebras
Time: 11:00 MWF
Instructor: Paul Terwilliger
Prerequisite: A good understanding of undergraduate linear algebra.
The course material should be accessible for first year graduate students.
Textbook: Quantum Groups, by Christian Kassel.
Graduate Texts in Mathematics Vol. 155. Springer Verlag, 1994.
DESCRIPTION: Quantum groups were introduced during the 1980's in the study
of statistical mechanical models. Since then they have appeared in
many contexts, such as the study of knots and links, representation
theory, algebraic combinatorics, and special functions.
In this introductory course we will discuss the basic concepts
associated with quantum groups. Broadly speaking we will follow
the above text. Topics include:
The language of Hopf algebras
The quantum plane and its symmetries
The Lie algebra sl2 and quantum algebra Uq(sl2)
The Hopf algebra structure for Uq(sl2)
The Yang-Baxter equation
Drinfeld's quantum double
Knots and links
The Jones-Conway polynomial
This course is recommended for anyone interested in
Lie theory, algebraic combinatorics, special functions,
knot invariants, and statistical mechanical models.
Notes: The following article gives a nice overview of how Hopf
algebras and quantum groups come up in representation
theory, symmetric functions, and the theory of partitions:
Darij Grinberg and Victor Reiner.
Hopf Algebras in Combinatorics. arXiv:1409.8356v1 30 Sep 2014.
The following article shows how Hopf algebras appear in
special functions, orthogonal polynomials, and hypergeometric series:
Martijn Caspers and Erik Koelink.
Quantum groups and Special Functions.
http://www.math.ru.nl/~koelink/edu/GQSF-Bizerte-2010.pdf
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ISBN-13: 978-0387943701 ISBN-10: 0387943706
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