Affine quantum groups of type A - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:14:04Z http://mathoverflow.net/feeds/question/75825 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75825/affine-quantum-groups-of-type-a Affine quantum groups of type A Clinton Boys 2011-09-19T07:04:47Z 2011-09-19T23:44:01Z <p>I have a general question regarding quantum groups. It seems to me that the representation theory of the algebra <code>$\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$</code> has many parallels with the representation theory of <code>$\mathfrak{sl}_{e-1}$</code>. Why is it this algebra which shares so much in common with <code>$\mathfrak{sl}_{e-1}$</code> rather than its universal enveloping algebra <code>$\mathcal{U}_q(\mathfrak{sl}_{e-1})$</code>? (<code>$\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$</code> is the quantised universal enveloping algebra of the Lie algebra <code>$\mathfrak{sl}_{e-1}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbf{C}c\oplus \mathbf{C}d$</code>). </p> <hr> <p><strong>Edit:</strong> Let me try to formulate the question more precisely. Let <code>$\mathfrak{g}=\mathfrak{sl}_{e-1}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbb{C}c\oplus \mathbb{C}d$</code> and let $U_q(\mathfrak{g})$ be its quantised universal enveloping algebra. Then the representation theory of $U_q(\mathfrak{g})$ has a lot in common with the representation theory of $\mathfrak{sl}_{e-1}$, for example there is a theory of integrable highest weight modules for the two objects and their finite-dimensional irreducible modules have a similar appearance. I am wondering why it is this algebra which shares properties with <code>$\mathfrak{sl}_{e-1}$</code>, rather than its own quantised universal enveloping algebra $U_q(\mathfrak{sl}_{e-1})$. If this doesn't make sense, then I mustn't understand something!</p> http://mathoverflow.net/questions/75825/affine-quantum-groups-of-type-a/75863#75863 Answer by Maarten Bergvelt for Affine quantum groups of type A Maarten Bergvelt 2011-09-19T16:05:00Z 2011-09-19T16:05:00Z <p>Hm, are you asking about the similarities between the representation theory of non quantum affine algebras ( <code>$\mathcal{U}(\widehat{\mathfrak{sl}}_{e-1})$</code>) and that of the quantum group <code>\$\mathcal{U}<em>q(\mathfrak{sl}</em>{e-1})$</code> corresponding to the finite dimensional simple Lie algebra?</p> <p>That similarity is part of a wider pattern, see the famous picture on the cover of the Etingof, Frenkel and Kirillov book on Representation theory: </p> <p><a href="http://books.google.com/books?hl=en&amp;lr=&amp;id=LrIpQIpvRzMC&amp;oi=fnd&amp;pg=PR13&amp;dq=etingof+representation+theory&amp;ots=g_V9CpeCYe&amp;sig=44-V8VxDAV6DFQSA8nE-zH4dCsg" rel="nofollow">http://books.google.com/books?hl=en&amp;lr=&amp;id=LrIpQIpvRzMC&amp;oi=fnd&amp;pg=PR13&amp;dq=etingof+representation+theory&amp;ots=g_V9CpeCYe&amp;sig=44-V8VxDAV6DFQSA8nE-zH4dCsg</a></p>