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Morita equivalence of Nijenhuis structures

Andrés I. Rodríguez

Abstract

We introduce Morita equivalence for Nijenhuis groupoids and for their infinitesimal counterparts, establishing a global-to-infinitesimal correspondence under the Lie functor. A special case is that of holomorphic Lie groupoids and algebroids. We use our framework to enhance the known Morita equivalences for quasi-symplectic groupoids and Dirac structures with compatible Nijenhuis structures.

Morita equivalence of Nijenhuis structures

Abstract

We introduce Morita equivalence for Nijenhuis groupoids and for their infinitesimal counterparts, establishing a global-to-infinitesimal correspondence under the Lie functor. A special case is that of holomorphic Lie groupoids and algebroids. We use our framework to enhance the known Morita equivalences for quasi-symplectic groupoids and Dirac structures with compatible Nijenhuis structures.
Paper Structure (28 sections, 24 theorems, 109 equations)

This paper contains 28 sections, 24 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Nijenhuis Groupoids
  3. Preliminaries on Nijenhuis structures
  4. Multiplicative Nijenhuis structures
  5. Morita equivalence of Nijenhuis groupoids
  6. Morita equivalence is an equivalence relation.
  7. Infinitesimal Nijenhuis structures
  8. Characterization of IM tensor fields via 1-derivations
  9. Correspondence between linear tensor fields and $1$-derivations.
  10. The Nijenhuis condition in terms of 1-derivations.
  11. Infinitesimally multiplicative 1-derivations.
  12. The Lie correspondence for (1,1)-tensor fields
  13. Integration of tangent lifts.
  14. Morita equivalence of infinitesimal structures
  15. The sum of the actions.
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $N\in \Omega^{1}(P,TP)$ be a $(1,1)$-tensor field and let $\varphi:P\to B$ be a surjective submersion. Then $N$ is $\varphi$-projectable if and only if $N\left( \operatorname{ker}\mathrm{d}\varphi \right) \subseteq \operatorname{ker}\mathrm{d}\varphi$ and the induced map $\bar{N}:TP/\operatornam

Theorems & Definitions (70)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 60 more