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Hello all,

I know basic representation theory(finite groups, lie groups&lie algebras) and I want to get a flavor of quantum groups (why they are useful, important results etc) and other related things like the Yang-Baxter equation. Can someone suggest me some good expository articles? Thank you.

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  • $\begingroup$ Related mathoverflow.net/questions/73261/… "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?" $\endgroup$
    – Alexander Chervov
    Commented Dec 5, 2012 at 10:16

9 Answers 9

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Drinfeld's original ICM-86 talk "Quantum groups" is something "must read", scanned files are available here.

This old introduction works out many details and is quite good: "An introduction to quantized Lie groups and algebras" T.Tjin arXiv:hep-th/9111043

There is certain interplay between certain topics in classical simple Lie algebras and quantum groups, in particular the Yangian. A. Molev's survey is quite good for this topic: Yangians and their applications http://arxiv.org/abs/math/0211288

Concerning the books let me be the second one on the Christian Kassel's book - it is good introduction in the series "Graduate texts in math" and it is probably one the best for beginners.

A Guide to Quantum Groups Vyjayanthi Chari , Andrew N. Pressley is one of the most comprehensive books

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Don't forget the "classics" on compact quantum groups:

Woronowicz, S.L., Compact quantum groups. Symétries quantiques (Les Houches, 1995), 845–884, North-Holland, Amsterdam, 1998. See also http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf

Maes, Ann; Van Daele, Alfons, Notes on compact quantum groups. Nieuw Arch. Wisk. (4) 16 (1998), no. 1-2, 73–112. See also http://arxiv.org/abs/math/9803122

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Christian Kassel's book on the subject is a classic, and a great one.

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I had a related question regarding locally compact quantum groups sometime ago, but was not confident to ask it on MO. However, I asked Matthew Daws, and he recommended to me the following pretty nice introduction:

Locally compact quantum groups by J. Kustermans (2003).

Though maybe not exactly what you are looking for, this tutorial style chapter covers a lot of useful material.

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This is mostly meant as a reminder that the original question asks about articles and not books or research papers. Of the latter there are a huge number by now. The notion of "quantum group" has multiple aspects, not easily covered by a single exposition, but it's reasonable to look for a fairly brief guide to what is going on in the subject (and why). Some of the lecture notes and other expositions mentioned here should be useful, at least within their own defined limits, but the large books such as Chari-Pressley and the research monographs such as Lusztig's book go far beyond the scope of "article". (Jantzen's more introductory textbook on the other hand is a fairly elementary introduction to one important line of work, though not to all possible ones.)

There isn't actually a precise mathematical definition of "quantum group", which is definitely a problem with the kind of free-flowing discussion in the answers here. Hopf algebra theory is more narrowly defined, but even here there are too many directions to encompass in a survey article or set of lecture notes.

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These are a nice set of introductory notes that I like discussing the example of quantum $SL_{2},\mathfrak{sl}_{2}$.

Also, Kashiwara's original papers on the 'crystals' and 'crystal bases' in quantised universal enveloping algebras are very readable and discuss the relationship between the representation theory of these objects and Kac-Moody algebras. Essentially, for generic $q$ the representation theories are the same. Jantzen's AMS book goes into further detail on this story and has a whole chapter devoted to several examples highlighting some of the main features.

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You might enjoy the short book by Ross Street: Quantum groups: a path to current algebra (Cambridge University Press, 2007).

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Verdier, Jean-Louis

Groupes quantiques

Séminaire Bourbaki, 29 (1986-1987), Exposé No. 685, 15 p. numdam.org

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Concerning compact and locally compact quantum groups, I recommend T.Timmermann's book "An invitation to quantum groups and duality".

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