Dirac pairs

TL;DR

This paper generalizes the concept of Nijenhuis torsion to relations on Lie algebroids, proves the torsion vanishes for bi-Hamiltonian structures, and introduces Dirac pairs linking compatible structures on manifolds and Lie algebroids.

Contribution

It extends Nijenhuis torsion to relations, proves torsion vanishes for bi-Hamiltonian structures, and introduces Dirac pairs connecting various compatible structures.

Findings

Torsion of relations in Lie algebroids can be defined and analyzed.

Torsion vanishes for relations defined by bi-Hamiltonian structures.

Dirac pairs relate different compatible geometric structures.

Abstract

We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion with the various kinds of compatible structures on manifolds, more generally on Lie algebroids.