  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
$(\mathcal{A},.)$, the orbits of the action $\Phi$ of $(\mathcal{A},+)$ on
$\mathcal{A}^*$ given by $\Phi(a,\mu)=\exp(L_a^*)(\mu)$ are pseudo-Hessian
manifolds, where $L_a(b)=a.b$. We illustrate this result by considering many
examples of associative commutative algebras an show that the pseudo-Hessian
manifolds obtained are very interesting.
