$F$-algebroids and deformation quantization via pre-Lie algebroids

TL;DR

This paper introduces a new framework for $F$-algebroids, explores their deformation theory via pre-Lie algebroids, and develops dualities and generalizations with applications to integrable systems.

Contribution

It generalizes $F$-algebroids through pre-Lie deformations, develops their cohomology, and introduces pre-$F$-algebroids with dualities and applications.

Findings

$F$-algebroids are semi-classical limits of pre-Lie deformations.

Deformation cohomology classifies infinitesimal and extended deformations.

New $F$-algebroids constructed via Nijenhuis operators.

Abstract

In this paper, first we introduce a new approach to the notion of $F$-algebroids, which is a generalization of $F$-manifold algebras and $F$-manifolds, and show that $F$-algebroids are the corresponding semi-classical 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 $F$-algebroids 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 and their applications to integral systems are discussed.