Smooth loops and loop bundles

TL;DR

This paper explores the properties of smooth loops, their tangent algebras, and introduces loop bundles, connecting them to known structures like $G_{2}$-structures and analyzing their geometric and algebraic features.

Contribution

It generalizes Lie groups to smooth loops, develops a tangent algebra framework, and introduces loop bundles with connections and torsion, linking to $G_{2}$-structures.

Findings

Established a loop analog of the Maurer-Cartan equation

Defined and analyzed torsion in loop bundles

Connected loop structures to $G_{2}$-geometry

Abstract

A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Mauer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also consider the critical points of some related functionals. Throughout, we see how some of the known properties of $G_{2}$-structures can be seen from this more general setting.