# Reference for Quantum groups

I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is particularly fond of a certain reference and why.

Also, I am very interested in ms/phd thesis (particularly ms thesis) that deal with quantum groups(I find that ms thesis sometimes aim to prove a few major theorems about a subject and fill in the details, instead of setting out to discover new mathematics). Thanks in advance.

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I have converted the question to community wiki, as it's asking for a big list of references, and there is no single right answer. –  Zev Chonoles Jul 23 '11 at 16:32
Christian Kassel's "Quantum Groups" is an excellent book. –  Grumpy Parsnip Jul 23 '11 at 16:35
Are you more interested in the approach that builds the general theory or rather some nice applications (there are tons in physics alone) with theory mentioned along the way? –  Marek Jul 23 '11 at 16:37
@Marek, I am not interested in the application as much as the general theory. –  Edison Jul 23 '11 at 16:39
@Marek, can you suggest some literature(for a non-physicist) that treats the physics first and the math along the way? Thanks. –  Edison Jul 25 '11 at 17:24

Our (Carter, Flath, Saito) "Classical and Quantum $6j$-Symbols," PUP does not cover the subject in general, but tells a lot of what is needed in the $U_q(sl_2)$ case.

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It's a more recent book, but I've read An introduction to quantum group and duality from Thomas Timmermann. I love how this book is written, although a lot of the proofs are missing. It's more of a year, year-and-a-half course. He closes the book with a treatment of Baaj-skandalis's beautiful result on duality of actions and coactions (which is my favorite topic) and then proceeds to extend results by means of very tough operator algebraic methods to the framework of quantum groupoids.

Works on this area of quantum groupoids is done by Gabriella böhm, ping Xu, Timmermann, Michael Enock, to cite a few. You can arxiv them to find a lot of their work. See also the notion of weak hopf algebras and hopf algebroids.

This "duality theory" they are trying to develop, outside the classical group-vector space framework, is a vast generalization of pontryagin's duality theorem, which states that the bidual of an abelian locally compact group is isomorphic to the said group and a later theory of tannaka-krein duality which evolved in order to remove the commutation hypothesis. These types of theorems ellucidate the representation theory of the algebraic structure, which in turn yields tons of amazing structural results on the algebraic structure itself. So this is a pretty nice thing to study. Remember that function analysis is, roughly speaking, the study of functionals on vector spaces, which are elements of the dual vector space.

Hope I could be of help.

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For starters you can check references in this question of mine on applications of Hopf algebras.

Here's an article or two that build a little theory of ${\mathfrak sl}_2$ related stuff. There are many similar articles on spin-chains where ${\mathfrak sl}_2$ deformations are naturally present.

References in the classic Di Francesco, Mathieu, Sénéchal on conformal field theory.

Another connection (via Yang-Baxter equation) is with integrable systems.

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Brain dump: Kassel is good. Jantzen (Lectures on Quantum Groups) is nicely written and clear, it's my favourite. Majid (Foundations of Quantum Group Theory) is an option, I haven't read it. Lusztig's book has a reputation of being tough to read. Chari and Pressley (A Guide To...) is comprehensive and has some interesting material on knots, 3-manifolds, the KZ equation and the absolute Galois group.

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