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

Alice Barbara Tumpach[1]

  • [1] Université Lille 1 Laboratoire Painlevé 59 655 Villeneuve d’Ascq Cedex (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 167-197
  • ISSN: 0373-0956

Abstract

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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 previously constructed by the author via a hyperkähler reduction.

How to cite

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Tumpach, Alice Barbara. "Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits." Annales de l’institut Fourier 59.1 (2009): 167-197. <http://eudml.org/doc/10389>.

@article{Tumpach2009,
abstract = {In this paper, we construct a hyperkähler structure on the complexification $\mathcal\{O\}^\mathbb\{C\}$ of any Hermitian symmetric affine coadjoint orbit $\mathcal\{O\}$ 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 $\mathcal\{O\}$. By a relevant identification of the complex orbit $\mathcal\{O\}^\mathbb\{C\}$ with the cotangent space $T\mathcal\{O\}$ of $\mathcal\{O\}$ induced by Mostow’s decomposition theorem, this leads to the existence of a hyperkähler structure on $T\mathcal\{O\}$ compatible with Liouville’s complex symplectic form and whose restriction to the zero section is the Kähler structure of $\mathcal\{O\}$. 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 previously constructed by the author via a hyperkähler reduction.},
affiliation = {Université Lille 1 Laboratoire Painlevé 59 655 Villeneuve d’Ascq Cedex (France)},
author = {Tumpach, Alice Barbara},
journal = {Annales de l’institut Fourier},
keywords = {Infinite-dimensional hyperkähler manifolds; affine coadjoint orbit; Hermitian-symmetric spaces; hyperkähler reduction; cotangent space; strongly orthogonal roots; $L^*$-algebra; restricted Grassmannian; infinite-dimensional hyper-Kähler manifolds; hyper-Kähler reduction; -algebra},
language = {eng},
number = {1},
pages = {167-197},
publisher = {Association des Annales de l’institut Fourier},
title = {Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits},
url = {http://eudml.org/doc/10389},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Tumpach, Alice Barbara
TI - Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 167
EP - 197
AB - In this paper, we construct a hyperkähler structure on the complexification $\mathcal{O}^\mathbb{C}$ of any Hermitian symmetric affine coadjoint orbit $\mathcal{O}$ 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 $\mathcal{O}$. By a relevant identification of the complex orbit $\mathcal{O}^\mathbb{C}$ with the cotangent space $T\mathcal{O}$ of $\mathcal{O}$ induced by Mostow’s decomposition theorem, this leads to the existence of a hyperkähler structure on $T\mathcal{O}$ compatible with Liouville’s complex symplectic form and whose restriction to the zero section is the Kähler structure of $\mathcal{O}$. 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 previously constructed by the author via a hyperkähler reduction.
LA - eng
KW - Infinite-dimensional hyperkähler manifolds; affine coadjoint orbit; Hermitian-symmetric spaces; hyperkähler reduction; cotangent space; strongly orthogonal roots; $L^*$-algebra; restricted Grassmannian; infinite-dimensional hyper-Kähler manifolds; hyper-Kähler reduction; -algebra
UR - http://eudml.org/doc/10389
ER -

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