Wrap groups of fiber bundles over quaternions and octonions

TL;DR

This paper investigates wrap groups of fiber bundles over real, complex, quaternion, and octonion fields, establishing their infinite-dimensional Lie group structures and manifold properties, while noting they may not satisfy the Campbell-Hausdorff formula.

Contribution

It introduces the construction of wrap groups over various fields, proves their Lie group and manifold structures, and explores their algebraic properties across different number systems.

Findings

Wrap groups form infinite-dimensional Lie groups for differentiable fibers.

These groups have manifold structures aligned with the underlying field (real, complex, quaternion, octonion).

They may not satisfy the Campbell-Hausdorff formula locally.

Abstract

This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition $(f,g)\mapsto f^{-1}g$ is continuous or differentiable depending on a class of smoothness of groups. Moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. Nevertheless, it is proved that these groups does not necessarily satisfy the Campbell-Hausdorff formula even locally.