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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?

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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 Crooks
    Commented Jan 10, 2014 at 23:26
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    $\begingroup$ 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)? $\endgroup$
    – YCor
    Commented Jan 10, 2014 at 23:45
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    $\begingroup$ 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 non-affine loop group, so the notation is a bit confusing. When you say "equivariant", do you refer to the adjoint action? $\endgroup$
    – S. Carnahan ♦
    Commented Jan 11, 2014 at 1:55

1 Answer 1

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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.

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  • $\begingroup$ This book looks like something I needed a year ago. Thanks! $\endgroup$
    – Sven Cattell
    Commented Jan 14, 2014 at 16:53
  • $\begingroup$ Is there any reference that shows that the smooth loop group is regular? - nm, found it $\endgroup$
    – Sven Cattell
    Commented Jan 15, 2014 at 17:03

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