For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. Is it possible to lift this map to an equivariant map on the level of groups? How is it done?



Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?
Use theorem 40.3 of 


* Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997.[(pdf)](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf)


You will have to handle some fundamental group obstructions. Equivariance (with respect to rotations of the parameter, I guess, or even with respect to the whole reparameterization group $Diff(S^1)$) is preserved. 


