### Abstract

The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10], [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm's equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of sl(2, ℂ).

Original language | English (US) |
---|---|

Pages (from-to) | 279-302 |

Number of pages | 24 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1997 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Annales Scientifiques de l'Ecole Normale Superieure*, vol. 30, no. 3, pp. 279-302. https://doi.org/10.1016/S0012-9593(97)89921-1

**Hyper-Lie Poisson structures.** / Xu, Ping.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Hyper-Lie Poisson structures

AU - Xu, Ping

PY - 1997/1/1

Y1 - 1997/1/1

N2 - The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10], [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm's equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of sl(2, ℂ).

AB - The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10], [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm's equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of sl(2, ℂ).

UR - http://www.scopus.com/inward/record.url?scp=0031536976&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031536976&partnerID=8YFLogxK

U2 - 10.1016/S0012-9593(97)89921-1

DO - 10.1016/S0012-9593(97)89921-1

M3 - Article

AN - SCOPUS:0031536976

VL - 30

SP - 279

EP - 302

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

SN - 0012-9593

IS - 3

ER -