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

John Alexander Cruz Morales, Jiefeng Liu and Yunhe Sheng

Vol. 326 (2023), No. 2, 251–284

DOI: 10.2140/pjm.2023.326.251

Abstract

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 $F$-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 are discussed.

Keywords

$F$-algebroids, pre-$F$-algebroids, eventual identity, Nijenhuis operator

Mathematical Subject Classification

Primary: 53D17, 53D45, 53D50, 53D55

Secondary: 14J33

Milestones

Received: 11 August 2022

Revised: 20 November 2023

Accepted: 22 November 2023

Published: 9 January 2024