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?



Hopf Algebras and Quantum Groups
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.


