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

Abstract

In this paper, we generalize all the results obtained on para-Kähler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-Kähler Lie algebroids. In particular, we study exact para-Kähler Lie algebroids as a generalization of exact para-Kähler 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 ${\mathcal{A}}^{\ast }$ given by $\Phi (a,\mu )=\exp (L_a^{\ast })(\mu )$ 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.

23 pages

References in corpus (3)

• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
• Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids
• Complete lists of low dimensional complex associative algebras

Cited by in corpus (2)

• Cohomologies and crossed modules for pre-Lie Rinehart algebras
• Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures