TY - JOUR

T1 - Hyper-Lie Poisson structures

AU - Xu, Ping

N1 - Funding Information:
Classijication. Primary 58 F 05. Secondary 53 B 35. by NSF grants DMS92-03398 and DMS95-04913.
Funding Information:
The author would like to thank Jorgen Andersen, Sean Bates, Olivier Biquard, Ranee Brylinski, Hansjijrg Geiges, Victor Cuillemin, Nigel Hitchin, Peter Kronheimer, Oscar ’ >arcia-Prada, Yvette Kosmann-Schwarzbach and Alan Weinstein for useful discussions and email correspondences. Special thanks go to Ludmil Katzarkov and Tony Pantev for calling his attention to the papers [lo], [I 11. Most of this work was carried out when the author was a member of MSRI, where the research was supported by NSF Grant DMS90-22140. In addition, the author wishes to thank the Center of Mathematics at Zhejiang University, China, for its hospitality while part of this work being done.

PY - 1997

Y1 - 1997

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).

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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 -