Fermionic Second Quantization and the Geometry of the Restricted Grassmannian

Wurzbacher, Tilmann

doi:10.1007/978-3-0348-8227-9_6

Cited by 36 publications

(47 citation statements)

References 51 publications

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“…for cAn Ã ; aAh Ã ; zAn; and ZAh: The requirement that this action preserve the preduals, together with properties (20) implies that jðZÞ Ã ðcÞCc and that oðZ; ÁÞ Ã À ðjðÁÞzÞ Ã ð Þ ð cÞCa for all ZAh and all zAn: Taking here alternatively Z ¼ 0 and z ¼ 0; the second condition becomes oðZ; ÁÞ Ã ðcÞCa and ðjðÁÞzÞ Ã ðcÞCa: We have proved the following theorem. …”

Section: Extensions Of Banach Lie-poisson Spacesmentioning

confidence: 96%

“…Let H be a complex separable Hilbert space endowed with a polarization[13,20], that is, a direct sum decomposition H ¼ H þ "H À into two closed orthogonal subspaces. Denote by P 7 : H-H 7 the orthogonal projectors on H 7 ; hence P þ þ P À ¼ id and P þ P À ¼ P À P þ ¼ 0: Denote by L N ðHÞ and L N ðH 7 Þ the Banach Lie algebra of bounded linear operators on H and H 7 ; respectively, relative to the commutator bracket.…”

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confidence: 99%

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Extensions of Banach Lie–Poisson spaces

Odzijewicz

Raţiu

2004

Journal of Functional Analysis

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Section: Extensions Of Banach Lie-poisson Spacesmentioning

confidence: 96%

mentioning

confidence: 99%

Extensions of Banach Lie–Poisson spaces

Odzijewicz

Raţiu

2004

Journal of Functional Analysis

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“…Let us denote byŨ res the central extension of U res with Lie algebraũ res and by p :Ũ res → U res the projection map in the exact sequence 1 → S 1 →Ũ res → U res → 1. The groupŨ res is isomorphic to the unitary subgroup of GL ∼ res , the central extension of the group of invertible elements in B res constructed in [33, Section 6.6] (see also [39,Section II.3]). The usual coadjoint action ofŨ res on the dual of its Lie algebra leaves the predual (ũ res ) * invariant since by Eq.…”

Section: The Banach Lie-poisson Space Associated To the Universal Cenmentioning

confidence: 99%

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

Beltiţă

Raţiu

Tumpach

2007

Journal of Functional Analysis

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We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach LiePoisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out.

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“…The Kähler structures on Gr 0 res (H + , H − ) obtained by these identifications are proportional to the standard one as defined in [27] or [38]. The complexified orbit O C D k is the set of operators on H with two distinct eigenvalues ik and −ik such that the corresponding eigenspaces P ik and P −ik belong respectively to Gr 0 res (H + , H − ) and Gr 0 res (H − , H + ).…”

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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Fermionic Second Quantization and the Geometry of the Restricted Grassmannian

Cited by 36 publications

References 51 publications

Extensions of Banach Lie–Poisson spaces

Extensions of Banach Lie–Poisson spaces

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

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

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