Nonassociative analogs of Lie groupoids

TL;DR

This paper introduces nonassociative geometric structures called quasiloopoids and loopoids, extending Lie groupoids, and explores their properties, tangent and cotangent bundles, and applications to discrete mechanics.

Contribution

It defines nonassociative analogs of Lie groupoids, proves tangent bundles of loopoids are loopoids, and connects these structures to skew-algebroids and discrete mechanics.

Findings

Tangent bundles of smooth loopoids are also loopoids.

Cotangent bundles of loopoids do not necessarily preserve the structure.

Nonassociative structures can model discrete mechanics similarly to Lie groupoids.

Abstract

We introduce nonassociative geometric objects generalising naturally Lie groupoids and called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids are canonically smooth loopoids again (it is nontrivial in the case of loopoids). We show also that this is not true if the cotangent bundles are concerned. After providing a few natural constructions, we show how the Lie-like functor associates with loopoids skew-algebroids and almost Lie algebroids and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.