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I have a general question regarding quantum groups. It seems to me that the representation theory of the algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$ has many parallels with the representation theory of $\mathfrak{sl}_{e-1}$. Why is it this algebra which shares so much in common with $\mathfrak{sl}_{e-1}$ rather than its universal enveloping algebra $\mathcal{U}_q(\mathfrak{sl}_{e-1})$? ($\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$ is the quantised universal enveloping algebra of the Lie algebra $\mathfrak{sl}_{e-1}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbf{C}c\oplus \mathbf{C}d$).


Edit: Let me try to formulate the question more precisely. Let $\mathfrak{g}=\mathfrak{sl}_{e-1}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbb{C}c\oplus \mathbb{C}d$ 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 $\mathfrak{sl}_{e-1}$, 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!

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    $\begingroup$ Clinton- I don't follow your question. The representation theory of $\mathfrak{sl}_e$ (as a Lie algebra) and that of $U(\mathfrak{sl}_e)$ (as an associative algebra) are the same thing (they are equivalent categories). For this question to make sense, you need to clarify what distinction you're making. $\endgroup$
    – Ben Webster
    Sep 19, 2011 at 15:32
  • $\begingroup$ Yes I understand that. My question is why is the representation theory of the universal enveloping algebra $U(\mathfrak{g})$ so similar to the representation theory of $\mathfrak{sl}_e$, where $\mathfrak{g}$ is the Lie algebra above. $\endgroup$ Sep 19, 2011 at 22:49
  • $\begingroup$ the quantised universal enveloping algebra* $\endgroup$ Sep 19, 2011 at 22:50
  • $\begingroup$ As Ben observes, the question needs more precise formulation. By now there has been a lot of study of relationships among module categories (or derived categories), which tends to get highly sophisticated but also interesting. See for example a 2004 JAMS paper and its references: ams.org/journals/jams/2004-17-03/S0894-0347-04-00454-0 Earlier Kazhdan-Lusztig worked out the subtle connection between certain representations of an affine Lie algebra and representations of a quantum group at a root of unity. Your type A case is combinatorially best behaved. $\endgroup$ Sep 19, 2011 at 23:10
  • $\begingroup$ Clinton- Your question is still pretty unclear and hard to answer. In general questions about why something is so are pretty hard to answer, but you also need to be a lot more specific about what facts you think need explaining. Lots of algebras have theories of highest weight modules. $\endgroup$
    – Ben Webster
    Sep 20, 2011 at 0:07

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Hm, are you asking about the similarities between the representation theory of non quantum affine algebras ( $\mathcal{U}(\widehat{\mathfrak{sl}}_{e-1})$) and that of the quantum group \$\mathcal{U}q(\mathfrak{sl}{e-1})$ corresponding to the finite dimensional simple Lie algebra?

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:

http://books.google.com/books?hl=en&lr=&id=LrIpQIpvRzMC&oi=fnd&pg=PR13&dq=etingof+representation+theory&ots=g_V9CpeCYe&sig=44-V8VxDAV6DFQSA8nE-zH4dCsg

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