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?
[Math] Hopf Algebras and Quantum Groups
hopf-algebrasqa.quantum-algebratextbook-recommendation
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These Hopf algebras are distinguished by the property that their representation categories admit a braided tensor structure. The space of deformations of the braided tensor structure on these particular categories is one dimensional (I believe this is a result of Drinfeld), so we get a universal family of braided deformations of U(g)-mod by varying q in U_q(g). The symmetric structure in U(g)-mod makes it a special point in this space, and one could argue that U(g)-mod is a "degeneration" of the generic braided behavior. There is unpublished work of Lurie on algebraic groups over the sphere spectrum that lends homotopy-theoretic support to this idea, since the symmetric structure doesn't manifest over the sphere.
Braided structures are important when studying topological (and conformal) field theories, since they describe the local behavior of embedded codimension 2 objects, such as points in a surface or links in a three-manifold. If you like homotopy theory, a braided tensor category is one that admits an action of the E[2] operad, whose spaces are (homotopy equivalent to) configuration spaces of points in the plane. Physically, these are the points where one inserts fields.
In principle, any statement about semisimple groups that can be phrased in braided-commutative (rather than fully commutative) language should be reconfigurable to a statement about these quantum groups. For example, there is a quantum local Langlands program (see the introduction of Gaitsgory's twisted Whittaker paper). Also, the representation theory of U_q(g) is interesting because of its connections to the representation theory of affine algebras and mod p representations (I think Kazhdan, Lusztig, and Bezrukavnikov are among the key names here).
The question is rather rambling and it is more about not so well-defined appetites (do you have a more conrete motivation?).
There is one thing which however makes full sense and deserves the consideration. Namely it has been asked what about higher categorical analogues of (noncommutative noncocommutative) Hopf algebras. This is not a trivial subject, because it is easier to do resolutions of operads than more general properads. Anyway the infinity-bialgebras are much easier than the Hopf counterpart. There is important work of Umble and Saneblidze in this direction (cf. arxiv/0709.3436). The motivating examples are however rather different than quantum groups, coming from rational homotopy theory, I think.
Similarly, there is no free Hopf algebra in an obvious sense what makes difficult to naturally interpret deformation complexes for Hopf algebras (there is a notion called free Hopf algebra, concerning something else). Boris Shoikhet, aided with some help from Kontsevich, as well as Martin Markl have looked into this.
Another relevant issue is to include various higher function algebras on higher categorical groups, enveloping algebras of higher Lie algebras (cf. baranovsky (pdf) or arxiv 0706.1396 version), usual quantum groups, examples of secondary Steenrod algebra of Bauese etc. into a unique natural higher Hopf setting. I have not seen that.
The author of the question might also be interested in a monoidal bicategorical approach to general Hopf algebroids by Street and Day.
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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:
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.