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 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.Comment: arXiv admin note: text overlap with arXiv:1412.499

On contact and symplectic Lie algebroids

“…Definition 2.1. A Lie algebroid A over a smooth manifold M is a vector bundle π : A → M together with a Lie algebra structure [ , ] on the space Γ(A) of sections and a bundle map ρ : A → T M called the anchor map such that (1) The induced map ρ : Γ(A) −→ X (M ) is a homomorphism of Lie algebras, that is, ρ([S 1 , S 2 ] A ) = [ρ(S 1 ), ρ(S 2 )] for S 1 , S 2 ∈ Γ(A). ( 2) For any sections S 1 , S 2 ∈ Γ(A) and for every smooth function f ∈ C ∞ (M ) the Leibniz identity [S 1 , f S 2 ] = f [S 1 , S 2 ] + (ρ(S 1 ) • f )S 2 is satisfied.…”

“…On the other hand, in the last decades, the Lie algebroids have an important place in the context of some different categories in differential geometry and mathematical physics and represent an active domain of research. ( [1], [2], [5], [9]) The Lie algebroids, are generalizations of Lie algebras and integrable distributions [10]. In fact a Lie algebroid is an anchored vector bundle with a Lie bracket on the module of sections and many geometrical notions which involves the tangent bundle were generalized to the context of Lie algebroids.…”