Hypersymplectic structures with torsion on Lie algebroids

Abstract

Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one correspondence between hypersymplectic structures with torsion and hyperkähler structures with torsion. We show that given a Lie algebroid with a hypersymplectic structure with torsion, the deformation of the Lie algebroid structure by any of the transition morphisms does not affect the hypersymplectic structure with torsion. We also show that if a triplet of $2$-forms is a hypersymplectic structure with torsion on a Lie algebroid $A$, then the triplet of the inverse bivectors is a hypersymplectic structure with torsion for a certain Lie algebroid structure on the dual $A^{\ast }$, and conversely. Examples of hypersymplectic structures with torsion are included.

arXiv admin note: text overlap with arXiv:1412.4992

References in corpus (5)

• Courant algebroids, derived brackets and even symplectic supermanifolds
• Geometry of Hyper-Kähler Connections with Torsion
• Hypercomplex structures on Courant algebroids
• Hyperstructures on Lie algebroids
• Hypersymplectic structures on Courant algebroids