Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 28 October 2012


Goal of this survey

The theory of quantum groups began its development in about 1982-1985. It is now 10 years since the 1986 ICM address of V.G. Drinfel'd ignited a wild frenzy of research activity in this area and things related to it. During this time quantum groups have become a "household" term in Lie theory in much the same way that Kac-Moody Lie algebras did in the 1970's. Given that quantum groups are now a part of every day Lie theory it seems desirable that there are treatments of the subject which are accessible to graduate students.

It has been my goal to produce a survey which is accessible to graduate students, and which contains the necessary background and the main results in the theory. I have chosen to make this a compendium of motivation, definitions and results. A secondary goal has been to write this in a relatively small space (long works are usually too daunting) and with this in mind I have chosen not to include any proofs. In many cases, providing a full proof would require introducing and developing some fairly sophisticated tools.

My main focus in these notes is to give a description of what the Drinfel'd-Jimbo quantum groups are, how one arrives at them and why they are natural. In the last chapter I shall explain how the Drinfel'd-Jimbo quantum groups are applied to get link invariants such as the Jones polynomial.

References for quantum groups

Drinfel'd's paper in the proceedings of the ICM 1986 is a dense summary of many of the amazing results that he had obtained. This paper still remains a basic reference.

  1. [Dri1987] V.G. Drinfeld, Quantum Groups, in Proceedings of the International Congress of Mathematics, A.M. Gleason ed., pp. 798-820, American Mathematical Society, Providence 1987.

Between 1987 and 1995 literally thousands of papers on quantum groups have been published. The book by V. Chari and A. Pressley which appeared in 1994 has 70 pages of references in minuscule type! Instead of wading through this mass of literature I have decided to only refer you to the books on quantum groups which have begun to appear recently, as follows:

  1. [CPr1994] V. Chari and A. Pressley, "A Guide to Quantum Groups", Cambridge University Press, Cambridge, 1994.
  2. [Jan1995] J. Jantzen, "Lectures on Quantum Groups", Graduate Studies in Mathematics Vol. 6, American Mathematical Society, 1995.
  3. [Jos1995] A. Joseph, "Quantum groups and their Primitive Ideals", Ergebnisse der Mathematik und ihrer Grenzgebiete; 3 Folge, Bd. 29, Springer-Verlag, New York-Berlin, 1995.
  4. [Kas1995] C. Kassel, "Quantum groups", Graduate Texts in Mathematics 155, Springer-Verlag, New York, 1995.
  5. [Lus1993] G. Lusztig, "Introduction to Quantum Groups", Progress in Mathematics 110, Birkhauser, Boston, 1993.
  6. [Maj1995] S. Majid, "Foundations of quantum group theory", Cambridge University Press, 1995
  7. [SSt1993] S. Shnider and S. Sternbeg, "Quantum groups: From Coalgebras to Drinfel'd Algebras", Graduate Texts in Mathematical Physics Vol. 2, International Press, Cambridge, MA 1993.

I recommend [CPr1994] for obtaining a basic understanding of what quantum groups are, where they came from, what the main results are, and what was known as of about the end of 1993. It contains only easy proofs and sketches of more involved proofs, very often referring the reader to the original papers for the full details of proofs. This book, however, is very useful for understanding what is going on. The recent book [Jan1995] is written specifically for graduate students. It has an excellent choice of topics, thorough descriptions of the motivations at each stage and detailed proofs. The book [SSt1993] treats the deformation theory aspect of quantum groups in detail and the book [Lus1993] is the only one that covers the connection between the quantum group and perverse sheaves.

Some missing topics and where to find them

There are many beautiful things in the theory of quantum groups that we won't even have time to mention. A few of these are:

  1. Canonical and crystal bases and the Littelmann path model for representations, see [Jos1995] Chapt. 5-6 and [Jan1995] Chapt. 9-11.
  2. Yangians, see [CPr1994] Chapt. 12.
  3. Quasi-Hopf algebras and twisting, see [CPr1994] Chapt. 16 and [SSt1993] Chapt. 8.
  4. The Knizhnik-Zamalodchikov equation and hypergeometric functions, see [CPr1994] Chapt. 16, [Kas1995] Chapt. 19 and [SSt1993] Chapt. 12.
  5. Lie bialgebras, Poisson Lie groups, and symplectic leaves, see [CPr1994] Chapt. 1.
  6. Representations at roots of unity and the connection to representations of algebraic groups over a finite field, see [CPr1994] Chapt. 11 and [AJS1994].
  7. The connection between representations of quantum groups at roots of unity and representations of affine Lie algebras at negative level, see [CPr1994] Chapt. 11 and Chapt. 16 and [KLu1993].

Further reference for the background topics

Chapters I-IV consist of background material needed for the material on quantum groups. These chapters are:

  1. Hopf algebras and braided tensor categories
  2. Lie algebras and enveloping algebras
  3. Deformations of Hopf algebras
  4. Perverse Sheaves

The following book contains a very nice up-to-date account of the theory of Hopf algebras, and it also includes some useful things on quantum groups.

  1. [Mon1992] S. Montomery, "Hopf Algebras and their Actions on Rings", Regional Conference Series in Mathematics 82, American Mathematical Society, 1992.

The book by Chari and Pressley [CPr1994] contains a nice introduction to monoidal categories and braided monoidal categories. The following little book is a beautiful summary of the main results in semisimple Lie theory.

  1. [Ser1987] J.-P. Serre, "Complex Semisimple Lie algebras", Springer-Verlag, New York, 1987.

Comprehensive accounts of the theory of Lie algebras and enveloping algebras can be found in Bourbaki and in the book by Dixmier.

  1. [Bou1972] N. Bourbaki, "Groupes et Algébres de Lie, Chapitres I-VIII", Masson, Paris, 1972.
  2. [Dix1994] J. Dixmier, "Enveloping algebras", Amer. Math. Soc. (1994); originally published in French by Gauthier-Villars, Paris 1974 and in English by North Holland, Amsterdam 1977.

The following are standard (and very useful) texts in Lie theory.

  1. [Hum1980] J. Humphreys, "Introduction to Lie algebras and representation theory", Graduate Texts in Mathematics 9, Springer-Verlag, New York-Berlin, (3rd printing) 1980.
  2. [Kac1983] V. Kac, "Infinite dimensional Lie algebras", Birkhauser, Boston, 1983.

The most comprehensive reference for modern deformation theory, especially in regard to deformations of Hopf algebras, is the book by Shnider and Sternberg [SSt1993] listed above. The book [CPr1994] also contains a very informative chapter on deformation theory.

Unfortunately, to my knowledge, there is no good introductory text on the theory of perverse sheaves. The classical reference is the following monograph.

  1. [BBD1982] A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Astérisque 100 (1982), Soc. Math. France.

On the other hand, much of the background material to perverse sheaves, such as homological algebra and sheaf theory is classical and appears in many books. The first few chapters of the following book contain an introduction to these topics.

  1. [KSc1980] M. Kashiwara and P. Schapira, "Sheaves on Manifolds", Frundlehren der mathematischen Wissenschaften 292, Springer-Verlag, New York-Berlin, 1980.

On reading these notes

I advise the reader to begin immediately with Chapter V and find out what a quantum group is. One can always peek back at the earlier chapters and find out the definitions later. This makes it more fun and provides good motivation for learning the earlier background material. It also avoids getting bogged down before one even gets to the quantum group.

In a number of places I have chosen to make these notes "nonlinear". There have been some occasions when I have decided to repeat some definition or some statement. Also in a few places, I have used some terms and notations that have not been defined yet, with an appropriate reference to the place later in the text where the definitions and notations can be found. I have done this with the intention of making each section a somewhat complete set of ideas without disrupting any particular section with a myriad of lengthy definitions. Even though we may wish it so, ideas in mathematics are not really linear and this has been reflected in these notes. The reader should feel free to skip around in the notes whenever the inclination arises.

I have included a complete table of contents in the hope that it will be helpful to the reader as a tool for finding definitions and for organizing and motivating the structures. For the same reason I have given every small section a title. This way the reader can follow the process of the development, as well as the details. Think of the table of contents as a flow chart for the mathematics.


Even though the theory of quantum groups is less than 15 years old I shall not undertake the complicated task of giving appropriate references and credits concerning the sources of the theorems and their first proofs. I refer the reader to the above books on quantum groups for this information.

Let me stress that none of the theorems state in this manuscript are due to me with two possible exceptions. Chapt. I Proposition (5.5) and Chapt. VII Theorem (5.2) are more general than I know of in the existing literature. Chapt I. Proposition (5.5) is well known in the context of the quantum group and I am only pointing out here that the well known proof, see [Tan1992] Prop. 2.2.1, works for any quantum double. Chapt. VII Theorem (5.2) is a nontrivial, but very natural, extension of well known results which appear, for example, in [Jan1995] Chapt. 8. The crucial part of the proof is similar to the proof of [Jan1995] Lemma 8.3.

I have tried to indicate, at the beginning of each chapter, where one can find proofs of the theorems stated in that chapter. In many instances I have had to make minor changes in notations and statements in order to be consistent with the definitions that I have given. Especially since I have not included proofs the reader should be watchful and open to the possibility that there may be some minor errors.


First and foremost I thank Hans Wenzl for introducing me to the world of quantum groups and encouraging me to pursue research in related topics. He taught me the basics of quantum group theory and notes from a course he gave at University of California, San Diego have been tremendously useful over the years. I thank all of the audience members in my course in quantum groups at University of Wisconsin during Spring of 1994 for their interest, their suggestions and for coming so early in the morning.

I thank Gus Lehrer for inviting me to Australia, for making my year there a wonderful one and for suggesting my name for various invitations via which these notes have come into being. I thank Chuck Miller and John Cossey for the invitation to give a series of lectures on quantum groups to graduate students at the Workshop on Algebra, Geometry and Topology at Australian National University in Canberra during January 1996. These notes are an expanded version of the notes I distributed there. I thank Michael Murray and Alan Carey for inviting me to speak at the Australian Lie Groups Conference 1996 and for inviting me to contribute to these proceedings. Finally, I thank Dave Benson for some very helpful proofreading.

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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