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Lectures on Quantum Groups
Jens Carsten Jantzen, Aarhus University, Denmark

Graduate Studies in Mathematics
1996; 266 pp; hardcover
Volume: 6
Reprint/Revision History:
reprinted 1997
ISBN-10: 0-8218-0478-2
ISBN-13: 978-0-8218-0478-0
List Price: US$56
Member Price: US$44.80
Order Code: GSM/6
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See also:

Quantum Bounded Symmetric Domains - Leonid L Vaksman

Finite Dimensional Algebras and Quantum Groups - Bangming Deng, Jie Du, Brian Parshall and Jianpan Wang

Representations of Semisimple Lie Algebras in the BGG Category \(\mathscr {O}\) - James E Humphreys

Since its origin about ten years ago, the theory of quantum groups has become one of the most fascinating topics of modern mathematics, with numerous applications to several sometimes rather disparate areas, including low-dimensional topology and mathematical physics. This book is one of the first expositions that is specifically directed to students who have no previous knowledge of the subject. The only prerequisite, in addition to standard linear algebra, is some acquaintance with the classical theory of complex semisimple Lie algebras. Starting with the quantum analog of \(\mathfrak{sl}_2\), the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebras. The choice of the topics and the style of exposition make Jantzen's book an excellent textbook for a one-semester course on quantum groups.


Graduate students, research mathematicians, and theoretical physicists.


"Very useful for ... understanding and ... research in quantum groups, in particular, the chapters on the braid group action and crystal bases ... highly recommend[ed] ... to all research mathematicians working in quantum groups ... The writing is one of the most pleasant attributes of this book. The flow of the words and ideas is very smooth and very conducive to actually understanding what is going on."

-- Mathematical Reviews

"Carefully written ... there is an agreeable, sometimes informal, spirit with which [the author] tries to indicate to the reader what is really happening. One pleasant feature is his placing of long computations in appendices at the end of certain chapters."

-- Zentralblatt MATH

"The material is very well motivated ... Of the various monographs available on quantum groups, this one ... seems the most suitable for most mathematicians new to the subject ... will also be appreciated by a lot of those with considerably more experience."

-- Bulletin of the London Mathematical Society

Table of Contents

  • Introduction
  • Gaussian binomial coefficients
  • The quantized enveloping algebra \(U_{q}({\mathfrak sl}_2)\)
  • Representations of \(U_{q}({\mathfrak sl}_2)\)
  • Tensor products or: \(U_{q}({\mathfrak sl}_2)\) as a Hopf algebra
  • The quantized enveloping algebra \(U{_q}({\mathfrak g})\)
  • Representations of \(U{_q}({\mathfrak g})\)
  • Examples of representations
  • The center and bilinear forms
  • \(R\)-matrices and \(k_{q}[G]\)
  • Braid group actions and PBW type basis
  • Proof of proposition 8.28
  • Crystal bases I
  • Crystal bases II
  • Crystal bases III
  • References
  • Notations
  • Index
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