2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152135The 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 $\frak{sl}(2, \complex)$.LaTex, 26 pages, to appear in Ann. Sci. Ecole Norm. SupDifferential Geometry58F05 (Primary), 53B35 (Secondary)text