Kyoto Journal of Mathematics

Projective unitary representations of infinite-dimensional Lie groups

Bas Janssens and Karl-Hermann Neeb

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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.

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

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

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Zentralblatt MATH identifier

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]

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


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.

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