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Since I got a very good answer to my previous question, 217585, I am asking a sequel by moving up a level.

Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and its category of finite dimensional representations, $Y(\mathfrak{g})-\mathrm{mod}$.

We also have the affine Lie algebra $\widehat{\mathfrak{g}}$, its derived algebra $\widehat{\mathfrak{g}}'$ and the quantised enveloping algebra $U_q(\widehat{\mathfrak{g}}')$ and its category of finite dimensional representations, $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$.

Then my question is whether $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$ can be obtained from $Y(\mathfrak{g})-\mathrm{mod}$ by deformation quantisation? The Yangian, and therefore $Y(\mathfrak{g})-\mathrm{mod}$, is canonically constructed from $\mathfrak{g}$ and a related question is whether this holds for $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$?

This looks plausible for the categories as $Y(\mathfrak{g})-\mathrm{mod}$ gives rational solutions of the Yang-Baxter equation and $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$ gives trigonometric solutions of the Yang-Baxter equation. However it looks implausible for the Hopf algebras as $Y(\mathfrak{g})$ is based on $\mathfrak{g}[t]$ and $U_q(\widehat{\mathfrak{g}}')$ on $\mathfrak{g}[t,t^{-1}]$.

Note that this question is not answered by my previous question as the tensor product in $Y(\mathfrak{g})-\mathrm{mod}$ is not symmetric.

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  • $\begingroup$ This question is above my pay grade, so I'll just leave a comment. The parameter $t$ in the Yangian and the parameter $t$ in the quantum affine algebra are not supposed to be the same: rather, they should be related by the exponentiation map $\mathbb A^1 \to \mathbb G_m$ (which, to emphasize, is not an algebraic map). So the connection you're looking for will be best behaved not for the whole Yangian / quantum affine algebra, but just for the Yangian over $\mathbb K[[t]]$ and the quantum affine algebra over $\mathbb K[[t-1]]$. $\endgroup$ Sep 7 '15 at 16:02
  • $\begingroup$ That said, I believe there are deep connections between whole of $Y\mathfrak g$-mod and $U_q\hat{\mathfrak g}$-mod, but again the best connections involve exponentiating the spectral parameters on all modules. $\endgroup$ Sep 7 '15 at 16:03

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