Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

TL;DR

This paper introduces Nijenhuis forms on Lie-infinity algebras, providing a unified framework to understand deformations of Lie algebroids, Poisson-Nijenhuis structures, and related geometric structures.

Contribution

It offers a novel Nijenhuis interpretation of Lie algebroid deformations and demonstrates how Nijenhuis forms simplify the understanding of complex geometric structures.

Findings

Nijenhuis forms characterize deformations of Lie algebroids into Lie-infinity algebras.

Nijenhuis forms provide an efficient way to understand Poisson-Nijenhuis structures.

The approach unifies various geometric structures under a common Nijenhuis framework.

Abstract

Introducing Nijenhuis forms on Lie-infinity algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into a Lie-infinity algebra. Then we show that Nijenhuis forms on Lie-infinity algebras also give a short and efficient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.