On Hom-F-manifold algebras and quantization
Abdelkader Ben Hassin, Taoufik Chtioui, Mohamed Ali Maalaoui, Sami Mabrouk
Abstract
The notion of a F-manifold algebras is an algebraic description of a F-manifold. In this paper, we introduce the notion of Hom-F-manifold algebras which is generalisation of F-manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F-manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-F-manifold algebras which give rise to a Hom-F-manifold algebra through the sub-adjacent commutative Hom-associative algebra and the sub-adjacent Hom-Lie algebra. Using \huaO-operators on a Hom-F-manifold algebras we construct a Hom-pre-F-manifold algebras on a module. Then, we study Hom-pre-Lie formal deformations of commutative Hom-associative algebra and prove that Hom-F-manifold algebras are the corresponding semi-classical limits. Finally, we study Hom-Lie infinitesimal deformations and extension of Hom-pre-Lie n-deformation to Hom-pre-Lie (n+1)-deformation of a commutative Hom-associative algebra via cohomology theory.