2016/10/30 by Saïd Benayadi, Benayadi, Saïd, Mohamed Boucetta +1
Mathematics · #13P25 #53A15 #53C15 #53D17 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:13P25 #msc:53A15 #msc:53C15 #msc:53D17
paper · pdf · doi:10.48550/arxiv.1610.09682
23 pages
arxiv created 2016/10/30 · arxiv updated 2016/11/01
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 (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 an show that the pseudo-Hessian manifolds obtained are very interesting.