Projective unitary representations of infinite dimensional Lie groups

TL;DR

This paper establishes a correspondence between smooth projective unitary representations of infinite dimensional Lie groups and linear unitary representations of their extensions, providing a framework for understanding representations in infinite dimensions.

Contribution

It introduces a smooth structure on Lie group extensions and characterizes when Lie algebra representations integrate to the group, advancing the theory of infinite dimensional Lie group representations.

Findings

Every smooth projective representation corresponds to a linear representation of an extended group.

Characterization of Lie algebra representations that integrate to the group.

A precise correspondence between representations of the group, extension, and algebra.

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