On para-Kähler Lie algebroids and contravariant pseudo-Hessian structures

In this paper, we generalize all the results obtained on para‐Kahler Lie algebras in [3] to para‐Kahler Lie algebroids. In particular, we study exact para‐Kahler Lie algebroids as a generalization of exact para‐Kahler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra (A,.), the orbits of the action Φ of (A,+) on A∗ given by Φ(a,μ)=exp(La∗)(μ) are pseudo‐Hessian manifolds, where La(b)=a.b. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.