Structure "Hyper-Lie Poisson"

TL;DR

This paper explores hyperkahler and hypersymplectic structures within symplectic geometry, introducing hyper-Lie Poisson structures linked to Lie algebras, and demonstrates their application through Nahm's equations and moduli spaces.

Contribution

It defines hyper-Lie Poisson structures associated with semi-simple Lie algebras and provides criteria for their existence, extending the understanding of hypersymplectic geometry.

Findings

Hyper-Lie Poisson structures can be constructed for semi-simple Lie algebras.

Moduli spaces of Nahm's equations are realized as hypersymplectic leaves.

Connections between hypersymplectic structures and (co)adjoint orbits are established.

Abstract

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 $\frak{sl}(2, \complex)$.