most recent 30 from http://mathoverflow.net 2013-05-24T11:38:33Z http://mathoverflow.net/feeds/question/115231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups Qiao 2012-12-03T02:26:25Z 2013-01-05T23:44:51Z

<p>Hello all, </p> <p>I know basic representation theory(finite groups, lie groups&amp;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.</p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/115233#115233 George Melvin 2012-12-03T02:42:40Z 2012-12-03T02:42:40Z

<p><a href="http://www.mate.uncor.edu/~ggarcia/encuentros/notas-curso-qg-ha.pdf" rel="nofollow">These</a> are a nice set of introductory notes that I like discussing the example of quantum $SL_{2},\mathfrak{sl}_{2}$.</p> <p>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.</p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/115235#115235 Tom Leinster 2012-12-03T02:49:01Z 2012-12-03T02:49:01Z

<p>You might enjoy the short book by Ross Street: <a href="http://books.google.co.uk/books?id=u0slIpdBjYUC&amp;printsec=frontcover#v=onepage&amp;q&amp;f=false" rel="nofollow">Quantum groups: a path to current algebra</a> (<a href="http://www.cambridge.org/gb/knowledge/isbn/item1163997/Quantum%20Groups/?site_locale=en_GB" rel="nofollow">Cambridge University Press</a>, 2007).</p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/115236#115236 Mariano Suárez-Alvarez 2012-12-03T02:52:36Z 2012-12-03T02:52:36Z

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

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/115241#115241 Chandan Singh Dalawat 2012-12-03T03:37:19Z 2012-12-03T03:37:19Z

<p>Verdier, Jean-Louis</p> <p><em>Groupes quantiques</em> </p> <p>Séminaire Bourbaki, 29 (1986-1987), Exposé No. 685, 15 p. <a href="http://www.numdam.org/item?id=SB_1986-1987__29__305_0" rel="nofollow">numdam.org</a></p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/115252#115252 Alexander Chervov 2012-12-03T06:39:23Z 2012-12-03T06:39:23Z

<p>Drinfeld's original ICM-86 talk "Quantum groups" is something "must read", scanned files are available <a href="http://ncatlab.org/nlab/show/quantum+Yang-Baxter+matrix" rel="nofollow">here</a>.</p> <p>This old introduction works out many details and is quite good: "An introduction to quantized Lie groups and algebras" T.Tjin <a href="http://arxiv.org/abs/hep-th/9111043" rel="nofollow">arXiv:hep-th/9111043</a></p> <p>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 <a href="http://arxiv.org/abs/math/0211288" rel="nofollow">http://arxiv.org/abs/math/0211288</a></p> <p>Concerning the books let me be the second one on the <a href="http://www.amazon.com/Quantum-Groups-Graduate-Texts-Mathematics/dp/0387943706" rel="nofollow">Christian Kassel's book</a> - it is good introduction in the series "Graduate texts in math" and it is probably one the best for beginners.</p> <p><a href="http://www.amazon.com/Guide-Quantum-Groups-Vyjayanthi-Chari/dp/0521558840/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1354515722&amp;sr=1-1&amp;keywords=chari+quantum+groups" rel="nofollow">A Guide to Quantum Groups</a> Vyjayanthi Chari , Andrew N. Pressley is one of the most comprehensive books </p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/116796#116796 S. Sra 2012-12-19T16:39:33Z 2012-12-19T16:39:33Z

<p>I had a related question regarding <em>locally compact quantum groups</em> 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:</p> <p><a href="http://stubber.math-inf.uni-greifswald.de/algebra/special/notes/root.ps" rel="nofollow">Locally compact quantum groups</a> by <em>J. Kustermans</em> (2003).</p> <p>Though maybe not exactly what you are looking for, this tutorial style chapter covers a lot of useful material.</p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/116804#116804 Alexei Pirkovskii 2012-12-19T17:54:05Z 2012-12-19T17:54:05Z

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

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/118127#118127 Uwe Franz 2013-01-05T14:09:00Z 2013-01-05T14:09:00Z

<p>Don't forget the "classics" on compact quantum groups:</p> <p>Woronowicz, S.L., Compact quantum groups. Symétries quantiques (Les Houches, 1995), 845–884, North-Holland, Amsterdam, 1998. See also <a href="http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf" rel="nofollow">http://www.fuw.edu.pl/~slworono/PDF-y/CQG3.pdf</a></p> <p>Maes, Ann; Van Daele, Alfons, Notes on compact quantum groups. Nieuw Arch. Wisk. (4) 16 (1998), no. 1-2, 73–112. See also <a href="http://arxiv.org/abs/math/9803122" rel="nofollow">http://arxiv.org/abs/math/9803122</a></p>

http://mathoverflow.net/questions/115231/expository-papers-related-to-quantum-groups/118161#118161 Jim Humphreys 2013-01-05T23:44:51Z 2013-01-05T23:44:51Z

<p>This is mostly meant as a reminder that the original question asks about <em>articles</em> 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.)</p> <p>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.</p>