# Embedding quantum affine algebras at different levels

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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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. –  Alexander Braverman Oct 9 '11 at 19:53
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). –  Tony Licata Oct 12 '11 at 19:00