Hyperstructures on Lie Algebroids

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 hyperkahler structures. We show that the hypersymplectic framework is very rich in already known compatible pairs of tensors such as Poisson–Nijenhuis, ΩN and PΩ structures.