On para-K\"ahler Lie algebroids and generalized pseudo-Hessian structures

In this paper, we generalize all the results obtained on para-K"ahler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-K"ahler Lie algebroids. In particular, we study exact para-K"ahler Lie algebroids as a generalization of exact para-K"ahler Lie algebras. This study leads to a natural generalization of pseudo-Hessian manifolds. Generalized 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 $(\mathcal{A},\mathrm{.})$, the orbits of the action $\mathrm{\Phi }$ of $(\mathcal{A},+)$ on ${\mathcal{A}}^{\ast }$ given by $\mathrm{\Phi }(a,\mu )=\exp (L_a^{\ast })(\mu )$ are pseudo-Hessian manifolds, where $L_a(b)=a\mathrm{.}b$. We illustrate this result by considering many examples of associative commutative algebras an show that the pseudo-Hessian manifolds obtained are very interesting.