Hypersymplectic structures with torsion on Lie algebroids

TL;DR

This paper explores hypersymplectic structures with torsion on Lie algebroids, revealing their relation to Nijenhuis morphisms, twisted Poisson structures, and hyperk"{a}hler structures, along with invariance under deformations.

Contribution

It establishes a correspondence between hypersymplectic structures with torsion and hyperk"{a}hler structures with torsion, and analyzes their behavior under Lie algebroid deformations.

Findings

Each hypersymplectic structure with torsion determines three Nijenhuis morphisms.

Such structures are equivalent to twisted Poisson structures from a contravariant perspective.

Deformations by transition morphisms do not alter hypersymplectic structures with torsion.

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 hyperk\"{a}hler 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 $A^*$, and conversely. Examples of hypersymplectic structures with torsion are included.