TITLE: What are Quantum Groups?
QUESTION [38 upvotes]: I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. I am utterly ignorant of the subject, so could someone help me out and tell me

What is a quantum group? I read the wikipedia page and this article, but I'd like to hear more opinions/interpretations to get a better idea.
Why is this subject interesting? Is it an active area of mathematics?
How do quantum groups arise from physics? Could you give me some examples?
What precedent mathematical knowledge do I need to attack the subject?

Moreover, the prof. told me to go look at a paper by Maulik and Okounkov. I don't know the title of the publication, only that it should be somehow related to the subject of quantum groups, be around 200 pages and probably fairly recent (but I could misremember or have confused something he told me about some other subject, the conversation was pretty long).
Thank you very much for your help!

P.S.$1$: Never mind for the article, I found it. Additional questions:

If somebody has read said article, could you explain more or less what does it treat?
What additional (in the sense of: more than in the 4th question above) knowledge would be needed to understand the article?


P.S.$2$: I have found this blog post giving a very fast motivation for the concept of quantum group (from a mathematician's point of view). Unfortunately, there are no examples from physics.

P.S.$3$: Looking around in the nLab to see what they have to say about Hopf algebras and quantum groups, I stumbled upon this report by Drinfeld. I've not had the time to read it yet. When I'll have, I'll edit again or maybe write my own answer to my questions. Meanwhile other opinions/answers/references are always welcome.

REPLY [18 votes]: Ok, a couple of months have passed since I posted this question, and I have begun this project on quantum groups. I am now in the condition of answering a couple of the questions I posed.

There are many different definitions of quantum groups, all of which are related somehow. A possible definition is: a (Drinfeld-Jimbo) quantum group is a $1$-parameter deformation of the universal enveloping algebra $U\mathfrak{g}$ of a Lie algebra $\mathfrak{g}$ which is always a Hopf algebra.
The subject is interesting because it relates to the subject of quantum integrable systems in quantum mechanics. There is an ongoing effort trying to geometrize the notion of quantum groups (as stated at page $28$ of the paper of Maulik and Okounkov).
See the wonderful and exhaustive answer by Will Orrick.
To learn quantum groups you will need some knowledge of physics (to have a motivation for the subject and maybe to get some intuition), Lie groups and Lie algebras, and representation theory.

For the questions added in the edits:

I have just begun to read the paper, maybe in a couple of months, when I'll have a clearer idea of what is happening, I will update this answer to give an overview. From what I have been told by my professor, it should be about finding a geometrical approach to quantum groups.
To understand the paper easily you will need a lot of algebraic geometry (you should at least know what a Nakajima variety is, and know stuff about Gromov-Witten invariants) and at least some K-theory. I have to admit I am struggling, but I am also learning a lot.


Edit: There are a few more subjects you will need to know to approach the paper and hope to undertand the majority of it. I would advise to take a look at:

Symplectic geometry (for example here, or any of the other references you can easily find online), in particular you should know at least what a moment map is.
Geometric Invariant Theory (GIT), which you can look up here.
Equivariant cohomology, if you have access to Springer through an university, Bott's introduction is really clear, else you can probably find something online.

I will update if I find other useful stuff.