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

TL;DR

This paper extends the theory of para-K"ahler Lie algebras to Lie algebroids, introduces generalized pseudo-Hessian structures, and provides methods to construct non-trivial examples using associative algebras.

Contribution

It generalizes results from Lie algebras to Lie algebroids and develops a framework for pseudo-Hessian structures with new construction techniques.

Findings

Generalization of para-K"ahler Lie algebras to Lie algebroids.

Introduction of generalized pseudo-Hessian manifolds.

Construction of pseudo-Hessian manifolds from associative algebras.

Abstract

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.