Hyperstructures on Lie algebroids

TL;DR

This paper introduces hypersymplectic structures on Lie algebroids, establishing a correspondence with hyperk"ahler structures and demonstrating their connection to known tensor pairs like Poisson-Nijenhuis.

Contribution

It extends hypersymplectic structures to Lie algebroids and links them with classical geometric structures, providing a unifying framework.

Findings

Established a 1-1 correspondence between hypersymplectic and hyperk"ahler structures.

Unified various known tensor pairs within the hypersymplectic framework.

Extended classical hypersymplectic results to the setting of Lie algebroids.

Abstract

We define hypersymplectic structures on Lie algebroids recovering, as particular cases, all the classical results and examples of hypersymplectic structures on manifolds. We prove a 1-1 correspondence theorem between hypersymplectic structures and (pseudo-)hyperk\"{a}hler structures. We show that the hypersymplectic framework is very rich in already known compatible pairs of tensors such as Poisson-Nijenhuis, $\Omega N$ and $P \Omega$ structures.