Towards a Lie theory of locally convex groups

Neeb, Karl-Hermann

doi:10.1007/s11537-006-0606-y

Cited by 221 publications

(362 citation statements)

References 242 publications

(217 reference statements)

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“…A consequence is that the Levi-Civita connection (equivalently, the geodesic equation) need not exist; however, if the Levi-Civita connection does exist, it is unique. Let G be a smooth possibly infinite dimensional regular Lie group; see [77] or [76,Section 38] for the notion used here, or [106] for a more general notion of Lie group. Let G × E → E be a smooth group action on E and assume that B := E/G is a manifold.…”

Section: Riemannian Submersionsmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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Section: Riemannian Submersionsmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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“…Now φ has a Taylor expansion in homogeneous polynomials (cf. [31]), and the only nonzero term is the one of degree n. Hence, the preceding theorem applies. given by A⊗ n ∼ = C ∞ (X n ) [29,Theorem 4] and the S n -action on this algebra is induced by the natural action of S n on X n , the algebra S n (A) consists of the smooth symmetric functions on X n .…”

Section: Multiplicative Charactersmentioning

confidence: 77%

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Janssens

Neeb

2014

Int Math Res Notices

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Let K → X be a smooth Lie algebra bundle over a σ -compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k ⊗ R A R , where A R is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P ) 0 of the group of smooth gauge transformations of a principal fiber bundle P → X with compact base and structure group, and the groups SU n (A) 0 with A a complete involutive commutative continuous inverse algebra.

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AbstractWe solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb [21]. Moreover, we prove a result by the author announced in [21], which answers a question posed in an earlier, unpublished version of the survey.

Classification: 22E65 (Primary) 58B10 (Secondary)

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confidence: 78%