Integrating Nijenhuis Structures

TL;DR

This paper explores the global integration of Nijenhuis operators on manifolds, showing how their associated Lie algebroids can be integrated into Lie groupoids with additional structure, bridging local and global geometric structures.

Contribution

It establishes a correspondence between integrable Nijenhuis operators and Lie groupoids with extra structure, extending the understanding of Nijenhuis structures from infinitesimal to global objects.

Findings

Lie algebroids of Nijenhuis operators can be integrated into Lie groupoids with additional structure

The integration process is reversible, linking local Nijenhuis data to global groupoid structures

Examples illustrate the theoretical integration results

Abstract

A Nijenhuis operator on a manifold $M$ is a $(1,1)$ tensor $\mathcal N$ whose Nijenhuis-torsion vanishes. A Nijenhuis operator $\mathcal N$ on $M$ determines a Lie algebroid structure $(TM)_{\mathcal N}$ on the tangent bundle $TM$. In this sense a Nijenhuis operator can be seen as an infinitesimal object. In this paper, we identify its "global counterpart". Namely, we show that when the Lie algebroid $(TM)_{\mathcal N}$ is integrable, then it integrates to a Lie groupoid equipped with appropriate additional structure responsible for $\mathcal N$, and viceversa, the Lie algebroid of a Lie groupoid equipped with such additional structure is of the type $(TM)_{\mathcal N}$ for some Nijenhuis operator $\mathcal N$. We illustrate our integration result in various examples.