I am currently reading a paper discussing the convexity of the image of moment maps for loop groups. In particular, if G is a compact Lie group and [itex] S^1 [/itex] is the circle, the paper defines the loops group to be the set of function [itex] f: S^1 \to G [/itex] of "Sobolev class [itex] H^1 [/itex]." Now in the traditional sense, being of Sobolev class [itex] H^1 [/itex] means that the function and its first derivative are both square integrable. The question is, what does this mean for functions with codomain a compact Lie group?(adsbygoogle = window.adsbygoogle || []).push({});

So for real-valued functions whose domain is a compact Lie group, we could use the Haar integral to talk about square integrability. In this case, [itex] S^1 [/itex] conveniently just happens to be a compact, connected, abelian Lie group, though I'm sure we could even avoid the Haar measure by identifying [itex] S^1 [/itex] with the appropriate circles in R or C. But if the image of the function lies in a group, what do we do then? I suppose that we could use Whitney embedding to embed [itex] G \subseteq \mathbb R^n [/itex], or Peter-Weyl to embed [itex] G \subseteq U(n) [/itex], but then I would need to choose a norm on these spaces.

Even if that is the correct procedure, the better question is "what functions then are not in this space?" Since G is compact, composition with any norm would ensure that continuous functions are bounded. But we also have that the derivative exists, implying that all such functions would be continuous by necessity (or are we imposing weak-derivative conditions on this space)? That is, would it not then be sufficient to restrict ourselves to continuous functions?

Any insight would be appreciated.

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# Sobolev class of loops to a compact lie Group

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