For $l$ a positive integer, an affine Lie algebra $\widehat{\mathfrak{g}}$ has a level $l$ embedding $\phi_l: \widehat{\mathfrak{g}} \longrightarrow \widehat{\mathfrak{g}}$ which takes $x\otimes t^k$ to $x\otimes t^{kl}$ and multiplies the central extension by $l$. Can the map $\phi_l$ be deformed to a map of quantum affine algebras?
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1$\begingroup$ Tony, are you interested in this map as a map of algebras, or map of Hopf algebras? If you want a map of Hopf algebras, then I pretty sure that I can prove it doesn't exist. $\endgroup$– Alexander BravermanCommented Oct 9, 2011 at 19:53
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$\begingroup$ Good question. I suppose I am interested in a map of Hopf algebras, so that would answer it. (I'm slightly confused because I'm thinking of the quantum affine Lie algebra in Drinfeld's new realization rather than the Kac-Moody realization, and I don't know what the coproduct is from this point of view.) On the other hand, if one starts with an embedding $\mathfrak{g}_1\rightarrow \mathfrak{g}_2$, then perhaps the induced map of affinizations does deform to a map of quantum affine algebras (as Hopf algebras). $\endgroup$– Tony LicataCommented Oct 12, 2011 at 19:00
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