F-ALGEBROIDS AND DEFORMATION QUANTIZATION VIA PRE-LIE ALGEBROIDS

First we introduce the notion of F-algebroids, which is a generalization of F-manifold algebras and F-manifolds, and show that F-algebroids are the corresponding semiclassical limits of pre-Lie formal deformations of commutative associative algebroids. Then we use the deformation cohomology of pre-Lie algebroids to study pre-Lie infinitesimal deformations and extension of pre-Lie n-deformations to pre-Lie (n+ 1)-deformations of a commutative associative algebroid. Next we develop the theory of Dubrovin's dualities of F-algebroids with eventual identities and use Nijenhuis operators on Falgebroids to construct new F-algebroids. Finally we introduce the notion of pre- F-algebroids, which is a generalization of F-manifolds with compatible flat connections. Dubrovin's dualities of pre- F-algebroids with eventual identities, Nijenhuis operators on pre- F-algebroids are discussed.