Projective unitary representations of infinite dimensional Lie groups

Abstract

For an infinite dimensional Lie group $G$ modelled 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 $G^{♯}$ of $G$. (The main point is the smooth structure on $G^{♯}$.) For infinite dimensional Lie groups $G$ which are 1-connected, regular, and modelled on a barrelled 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 $G^{♯}$, and the appropriate unitary representations of its Lie algebra ${\mathfrak{g}}^{♯}$.

47 pages

References in corpus (3)

• Lie groups over non-discrete topological fields
• A non-smooth continuous unitary representation of a Banach-Lie group
• Solutions to open problems in Neeb's recent survey on infinite-dimensional Lie groups

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