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?
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$\begingroup$ Do you mean to ask whether $\phi$ is the differential at the identity of a morphism $LG\rightarrow LG$? $\endgroup$– Peter CrooksJan 10, 2014 at 23:26

1$\begingroup$ Could you explain better what is $\tilde{L}G$, what is $\mathbb{T}$ and what is the action? "affine loop group" only yields 9 Google occurrences. also, $\phi$ is equivariant for which group action(s)? $\endgroup$– YCorJan 10, 2014 at 23:45

1$\begingroup$ Presumably, $\mathbb{T}$ is the energy circle, and $\tilde{L}G$ is the central extension of the loop group by the level circle. I'm going to guess $\phi$ is a Lie algebra homomorphism. I think $LG$ is usually used to denote the nonaffine loop group, so the notation is a bit confusing. When you say "equivariant", do you refer to the adjoint action? $\endgroup$– S. Carnahan ♦Jan 11, 2014 at 1:55
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1 Answer
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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)
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.

$\begingroup$ This book looks like something I needed a year ago. Thanks! $\endgroup$ Jan 14, 2014 at 16:53

$\begingroup$ Is there any reference that shows that the smooth loop group is regular?  nm, found it $\endgroup$ Jan 15, 2014 at 17:03