Kyoto Journal of Mathematics

Projective unitary representations of infinite-dimensional Lie groups

Bas Janssens and Karl-Hermann Neeb

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Abstract

For an infinite-dimensional Lie group G modeled on a locally convex Lie algebra 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 modeled on a barreled Lie algebra g, we characterize the unitary 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 g.

Article information

Source
Kyoto J. Math., Volume 59, Number 2 (2019), 293-341.

Dates
Received: 17 February 2016
Revised: 17 January 2017
Accepted: 15 February 2017
First available in Project Euclid: 2 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1554170605

Digital Object Identifier
doi:10.1215/21562261-2018-0016

Mathematical Reviews number (MathSciNet)
MR3960295

Zentralblatt MATH identifier
07080106

Subjects
Primary: 17B15: Representations, analytic theory
Secondary: 17B56: Cohomology of Lie (super)algebras 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B68: Virasoro and related algebras 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 22E60: Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx} 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 22E66: Analysis on and representations of infinite-dimensional Lie groups 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05]

Keywords
infinite-dimensional Lie groups infinite-dimensional Lie algebras unitary representation theory

Citation

Janssens, Bas; Neeb, Karl-Hermann. Projective unitary representations of infinite-dimensional Lie groups. Kyoto J. Math. 59 (2019), no. 2, 293--341. doi:10.1215/21562261-2018-0016. https://projecteuclid.org/euclid.kjm/1554170605


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