# Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups?

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?

• Do you mean to ask whether $\phi$ is the differential at the identity of a morphism $LG\rightarrow LG$? Jan 10, 2014 at 23:26
• 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)?
– YCor
Jan 10, 2014 at 23:45
• 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? Jan 11, 2014 at 1:55

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.