On differentiable vectors for representations of infinite dimensional Lie groups

Neeb, Karl-Hermann

doi:10.1016/j.jfa.2010.07.020

Cited by 57 publications

(46 citation statements)

References 43 publications

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“…For issues related to smooth vectors for representations of infinite dimensional Lie groups see [7]. We say that a strongly continuous representation α on the dense domain D integrates to a continuous unitary representation of G if there exists such a representation π with D ⊆ H ∞ and dπ| D = α.…”

Section: Resultsmentioning

confidence: 99%

Integrating representations of Banach–Lie algebras

Merigon

2011

Journal of Functional Analysis

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Section: Resultsmentioning

confidence: 99%

Integrating representations of Banach–Lie algebras

Merigon

2011

Journal of Functional Analysis

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“…Let us first fix some notation before we turn to representations of possibly infinite-dimensional Lie groups, cf. [Nee06,Nee10]). 1.8 Definition Let K be a (possibly infinite-dimensional) Lie group and (ρ, V ) be a representation of K on a locally convex vector space V (i.e.…”

Section: Representations Of Lie Groups and Groupoidsmentioning

confidence: 99%

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

Amiri

Schmeding

2019

Ann Glob Anal Geom

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The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self maps. Then representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and smooth representations. It is known that in the topological category, this correspondence can be reversed for certain topological groupoids. We extend this result to the smooth category under weaker assumptions on the groupoids.

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“…4.4.1.7 (see Warner [69]) is satisfied, and thus Remark (3) We can consider spaces W l of C l -vectors in H as well, l ≥ 1, obtaining representations σ l : G → Aut(W l ). See Goodman [17] or Neeb [49]. Consider the tensor product representations σ k,l :…”

Section: Smooth Fréchet Bundleà La Bruhatmentioning

confidence: 99%

Induced C*-Complexes in Metaplectic Geometry

Krýsl

2018

Commun. Math. Phys.

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For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of Kostant's symplectic spinor bundle. Defining a Hilbert C * -structure on this bundle for a suitable C * -algebra, we obtain an elliptic C * -complex in the sense of Mishchenko-Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert C * -modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles. * E-mail address: Svatopluk.Krysl@mff.cuni.cz † The author thanks for financial supports from the founding No. 17-01171S granted by the Czech Science Foundation. We thank to the anonymous reviewer for his comments and suggestions.[14]. In [66] and [61], C * -compact perturbations of the complexes' differentials are allowed. For the infinite rank bundles, one cannot expect that, in general, the space of harmonic elements represents the cohomology groups homeomorphically (quotient topology) and linearly (quotient projections). Therefore, it makes sense to find conditions when this happens. This was investigated by the author in the past years (see [35], [36] or an overview in [37]).In the connection to sheaf cohomology, Banach and Fréchet bundles are studied in the papers of Illusie [25] and Röhrl [57]. To present a sample of the broad context (foremost connected to C * -algebras, K-theory and homological algebra) in which such bundles are considered, let us mention the papers of Maeda, Rosenberg [43], Freed, Lott [16], Larraín-Hubach [39], the author [36] and Fathizadeh, Gabriel [12]. There are also works in which holomorphic Banach bundles and their sheaf cohomology groups are treated. See Lempert [40] and Kim [27]. A further reason for an investigation of these complexes might originate in ), which we do not touch here explicitly, although we give a modest interpretation of the introduced structures in the realm of quantum theory.The aim of our paper is to study elliptic complexes for the case of infinite rank Hilbert C * -bundles which are induced by Lie group representations, and to show as well how the theory of connections and appropriately generalized elliptic complexes apply in this situation. In order to make the considered situation not too general, we choose a specific Lie group representation (the Segal-Shale-Weil representation of the metaplectic group). We hope that the reader may follow the text easier. Our further aim is to find examples for the theory of Mishchenko and Fomenko ([14]) that would not be trivial or covered by the theory developed around the classical Atiyah-Singer index theorem, and to show how the Hodge theory for certain C * -bundles (derived in [35] and [36]) make us able to describe the cohomology of elliptic complexes on Hilbert bundles easily.Let (V, ω 0 ) be a symplectic vector space and L be a Lagrangian subspace of it. The Segal-Shale-Weil representation is a non-trivial representation of the connected double cover G...

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On differentiable vectors for representations of infinite dimensional Lie groups

Cited by 57 publications

References 43 publications

Integrating representations of Banach–Lie algebras

Integrating representations of Banach–Lie algebras

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

Induced C*-Complexes in Metaplectic Geometry

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