The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits

Beltiţă, Daniel; Raţiu, Tudor S.; Tumpach, Alice Barbara

doi:10.1016/j.jfa.2007.03.001

Cited by 33 publications

(34 citation statements)

References 30 publications

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“…In this particular framework, restricting the action to these classical groups U p (H), certain results can be found in [6,7,8,16].…”

Section: B P (H) = {A ∈ B(h) : T R((amentioning

confidence: 97%

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Finsler geometry and actions of the $p$-Schatten unitary groups

Andruchow¹,

Larotonda²,

Recht³

2009

Trans. Amer. Math. Soc.

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Abstract. Let p be an even positive integer and U p (H) the Banach-Lie group of unitary operators u which verify that u − 1 belongs to the p-Schatten ideal B p (H). Let O be a smooth manifold on which U p (H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of U p (H). Our main result establishes that for any pair of given initial conditionsthere exists a curve δ(t) = e tz · x in O, with z a skew-hermitian element in the p-Schatten class, such thatwhich remains minimal as long as t z p ≤ π/4. Moreover, δ is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups U p (H) and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu

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“…In this particular framework, restricting the action to these classical groups U p (H), certain results can be found in [6,7,8,16].…”

Section: B P (H) = {A ∈ B(h) : T R((amentioning

confidence: 97%

“…Unitary orbits of operators have been previously studied from a geometric point of view in [3,4,6,7,10,16,19]. In this particular framework, restricting the action to these classical groups U p (H), certain results can be found in [6,7,8,16].…”

Section: B P (H) = {A ∈ B(h) : T R((amentioning

confidence: 99%

Finsler geometry and actions of the $p$-Schatten unitary groups

Andruchow¹,

Larotonda²,

Recht³

2009

Trans. Amer. Math. Soc.

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“…The first action defined above is transitive on the set of partial isometries contained in V + I with initial projection equal to V * V , while the second action is transitive on the set of partial isometries contained in V + I such that its initial projection and V * V have null index as a pair of projections. This result is closely related to the characterization of the connected components of the restricted Grassmannian (see [6], [7]). …”

Section: Which We Call the Generalized I-stiefel Manifold Associated mentioning

confidence: 68%

Geometry of $\mathfrak {I}$-Stiefel manifolds

Chiumiento

2010

Proc. Amer. Math. Soc.

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Abstract. Let I be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and U(H) I the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. In this paper we study the geometry of the unitary orbitswhere V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + I, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V * V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + I and homogeneous reductive spaces of U(H) I and U(H) I × U(H) I respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(H) I (or U(H) I × U(H) I ) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.

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“…which is a simple L * -group of compact type (a geometrical proof of this fact is given in [5]). The manifold Gr 0 res (H + , H − ) can be identified with a family of affine adjoint orbits of the Lie algebra u 2 of U 2 .…”

Section: Annales De L'institut Fouriermentioning

confidence: 96%

“…In particular, the Cartan subalgebras of a compact semi-simple Lie group are conjugate. This is no longer true in the infinite-dimensional case (see [5]). Remark A.6.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits

Tumpach¹

2009

Ann. inst. Fourier

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In this paper, we construct a hyperkähler structure on the complexification O C of any Hermitian symmetric affine coadjoint orbit O of a semi-simple L * -group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kähler structure of O. By a relevant identification of the complex orbit O C with the cotangent space T ′ O of O induced by Mostow's decomposition theorem, this leads to the existence of a hyperkähler structure on T ′ O compatible with Liouville's complex symplectic form and whose restriction to the zero section is the Kähler structure of O. Explicit formulas of the metric in terms of the complex orbit and of the cotangent space are given. As a particular case, we obtain the one-parameter family of hyperkähler structures on a natural complexification of the restricted Grassmannian and on the cotangent space of the restricted Grassmannian constructed by hyperkähler reduction in [30] . RésuméDans cet article, nous construisons une métrique hyperkählerienne sur l'orbite complexifiée O C de toute orbite coadjointe affine hermitienne symétrique O d'un L * -groupe semi-simple de type compact, qui est compatible avec la forme symplectique complexe de Kirillov-Kostant-Souriau et qui se restreint en la structure kählérienne de O. Grâce à une identification pertinente de l'orbite complexifiée O C avec l'espace cotangent T ′ O de l'orbite de type compact O induite par le théorème de décomposition de Mostow, nous en déduisons l'existence d'une structure hyperkählérienne sur T ′ O compatible avec la forme symplectique complexe de Liouville et dont la restriction à la section nulle est la structure kählérienne de O. Des formules explicites de la métriques en termes de l'orbite complexifiée et de l'espace cotangent sont données. Comme cas particulier, nous retrouvons la famille à un paramètre de structures hyperkählériennes sur une complexification naturelle de la grassmannienne restreinte et sur l'espace cotangent de la grassmannienne restreinte obtenue par réduction hyperkählérienne en [30].

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The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits

Cited by 33 publications

References 30 publications

Finsler geometry and actions of the $p$-Schatten unitary groups

Finsler geometry and actions of the $p$-Schatten unitary groups

Geometry of $\mathfrak {I}$-Stiefel manifolds

Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits

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