Structure "Hyper-Lie Poisson"

Details Structure "Hyper-Lie Poisson" Structure "Hyper-Lie Poisson"

1996-05-19

arxiv.org/abs/dg-ga/9605007v1

Mathematics

Differential Geometry

LaTex, 26 pages, to appear in Ann. Sci. Ecole Norm. Sup

Scientific paper

The main purpose of the paper is to study hyperkahler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras, 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 $\frak{su}(2)$ are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of $\frak{sl}(2, \complex)$.

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