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I have studied graduate abstract algebra and would like to learn about Hopf algebras and quantum groups. What book or books would you recommend? Are there other subjects that I should learn first before I begin studying Hopf algebras and quantum groups?

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Why the "suppose"? Have you, or have you not? (Also, since educational systems can vary, what have your abstract algebra courses covered?) –  Yemon Choi Aug 20 '11 at 4:22
I did take abstract algebra. It covered Group theory, Rings, Field Theory, Galois Theory, classical Algebraic geometry, modules, vectors spaces. It was a year long course. –  Ahmed Roman Aug 20 '11 at 7:06
I fear that a year long course covering the above-mentioned fields is, at best, an introduction. What you definitely will need in Hopf algebra theory is a good grip on tensor products, at least over fields (at the very least, the exactness properties and their consequences). Also, knowledge of representation theory a la arxiv.org/abs/0901.0827 is of much use; many Hopf-algebraic theorems generalize known facts of representation theory and seem utterly devoid of motivation if you don't know the latter. –  darij grinberg Aug 20 '11 at 12:03
Related question: mathoverflow.net/questions/115231/… "expository papers related to quantum groups " –  Alexander Chervov Dec 5 '12 at 10:17

4 Answers 4

up vote 11 down vote accepted

I don't think that you really need to learn much more algebra before you start on Hopf algebras. As long as you know about groups, rings, etc, you should be fine. An abstract perspective on these things is useful; e.g. think about multiplication in an algebra $A$ as being a linear map $m : A \otimes A \to A$, and then associativity of multiplication as being a certain commutative diagram involving some $m$'s. This naturally leads to dualization, i.e. coalgebras, comultiplication, coassociativity, etc, and then Hopf algebras come right out of there by putting the algebra and coalgebra structures together and asking for some compatibility (and an antipode).

For the Drinfeld-Jimbo type quantum groups, it is helpful to know some Lie theory, especially the theory of finite-dimensional semisimple Lie algebras over the complex numbers. If you don't know that stuff, the definitions will probably not be that enlightening for you.

There are a lot of books on quantum groups by now. They have a lot of overlap, but each one has some stuff that the others don't. Here are some that I have looked at:

  • Quantum Groups and Their Representations, by Anatoli Klimyk and Konrad Schmudgen. They have a penchant for doing things in excruciating, unenlightening formulas, but this book is the first one that I learned quantum groups from, so it remains the most familiar to me. This one has a lot more about Hopf $*$-algebras than any of the others.
  • A Guide to Quantum Groups, by Vijayanthi Chari and Andrew Pressley. Has an approach based more on Poisson geometry and deformation quantization.
  • Foundations of Quantum Group Theory, by Shahn Majid. Goes into more detail on braided monoidal categories, braided Hopf algebras, reconstruction theorems (i.e. reconstructing a Hopf algebra from its category of representations) than most other books, although some of this is covered in Chari-Pressley.
  • Quantum Groups, by Christian Kassel. Focuses mainly on $U_q(\mathfrak{sl}_2)$ and $\mathcal{O}_q(SL_2)$, and does a lot of stuff with knot invariants coming from quantum groups.
  • Hopf Algebras and Their Actions on Rings, by Susan Montgomery. This one is more about the theory of Hopf algebras than about Drinfeld-Jimbo quantum groups.

There are some other ones which I know are out there, but I haven't read. These include Lectures on Algebraic Quantum Groups, by Ken Brown and Ken Goodearl, Lectures on Quantum Groups, by Jens Jantzen, Introduction to Quantum Groups, by George Lusztig, and Quantum Groups and Their Primitive Ideals, by Anthony Joseph. Having glanced a little bit at the last two in this list, I found both of them more difficult to read than the ones in my bulleted list above.

So, as you can see, there is a lot of choice available. I would advise you to check a few of them out of the library and just see which one you like the best.

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Etingof also has a book on quantum groups. I haven't spent much time with it but it seems good. –  Peter Samuelson Aug 20 '11 at 20:46

I agree with MTS comment that you do not need much to start with. Whether you need more or not will depend on your focus: which part of the theory are you interested in?

Personally I would reccomend Serre's Complex Semisimple Lie Algebras as a must since you most likely will end up fighting with Serre's relations.

Also, classic books on Hopf algebras are not to be forgotten; here the choice is between Abe and Sweedler, if I remember correctly the title is "hopf algebras" in both cases.

As for the three books about which MTS does not add comments let me say something:

Joseph's book requires a solid background on Lie algebras and their reps, otherwise it's almost impossible to understand its directions. With this background it's a very intriguing, though demanding, reading.

Brown-Goodearl is a great book to start with. It's built from lecture notes of a course and therefore "learning oriented". It's much oriented towards the more algebraic part of the theory. I would reccomend McConnell-Robson book on Noncommutative Noetherian Rings at hand...

Lusztig's book requires feeling at ease with various categorical issues: definitely not a first reading. It is also uite narrowly focused on some specific aspects. I would not put it in any list: if that is your direction at some point younwill be forced into it.

I do not know about Jantzen's book.

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"Algebras of functions on quantum groups, Part 1" By Leonid I. Korogodski, Yan S. Soibelman –  Charlie Frohman Aug 20 '11 at 12:23
Yes, but that is the less algebraically oriented of all... I am still waiting for Part II –  Nicola Ciccoli Aug 20 '11 at 12:55

I think it is also worth to mention the book "S. Dascalescu: Hopf algebras. An introduction" as a suitable textbook on the algebraic theory of Hopf algebras. However, since it is not dealing with quantum groups, it could be timely to use it together with some books mentioned by MTS on this subject.

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