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!

  • $\begingroup$ There is a bit of a problem with mathjax at the moment. Bear with us. $\endgroup$ – David Roberts Sep 19 '11 at 7:33
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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 '11 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. $\endgroup$ – Clinton Boys Sep 19 '11 at 22:49
  • $\begingroup$ the quantised universal enveloping algebra* $\endgroup$ – Clinton Boys Sep 19 '11 at 22:50

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:



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