Lie rackoids

TL;DR

This paper introduces Lie rackoids, a new differential geometric structure that generalizes Lie groupoids to Leibniz algebroids, with applications to integrating Courant algebroids and providing new examples.

Contribution

It defines Lie rackoids, establishes their relation to Leibniz algebroids, and constructs examples including an integration of the Dorfman bracket without cocycles.

Findings

Lie rackoids relate to Leibniz algebroids similarly as Lie groupoids relate to Lie algebroids.

The tangent algebroid of a Lie rackoid is a Leibniz algebroid.

Provides examples including a Lie rackoid integrating the Dorfman bracket.

Abstract

We define a new differential geometric structure, called Lie rackoid. It relates to Leibniz algebroids exactly as Lie groupoids relate to Lie algebroids. Its main ingredient is a selfdistributive product on the manifold of bisections of a smooth precategory. We show that the tangent algebroid of a Lie rackoid is a Leibniz algebroid and that Lie groupoids gives rise via conjugation to a Lie rackoid. Our main objective are large classes of examples, including a Lie rackoid integrating the Dorfman bracket without the cocycle term of the standard Courant algebroid.