Projective unitary representations of infinite-dimensional Lie groups Article Swipe
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· 2019
· Open Access
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· DOI: https://doi.org/10.1215/21562261-2018-0016
· OA: W1537543197
For an infinite-dimensional Lie group $G$ modeled on a locally convex Lie algebra ${\\mathfrak{g}}$ , we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{\\sharp}$ of $G$ . (The main point is the smooth structure on $G^{\\sharp}$ .) For infinite-dimensional Lie groups $G$ which are $1$ -connected, regular, and modeled on a barreled Lie algebra ${\\mathfrak{g}}$ , we characterize the unitary ${\\mathfrak{g}}$ -representations which integrate to $G$ . Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of $G$ , smooth linear unitary representations of $G^{\\sharp}$ , and the appropriate unitary representations of its Lie algebra ${\\mathfrak{g}}^{\\sharp}$ .
