On Lie semiheaps and ternary principal bundles

TL;DR

This paper introduces Lie semiheaps, explores their relation to Lie groups, and generalizes principal bundles to a ternary setting, expanding the mathematical framework of differential geometry.

Contribution

It defines Lie semiheaps, proves the existence of invariant vector fields, and extends the concept of principal bundles to ternary structures, generalizing heapification.

Findings

Lie semiheaps are smooth manifolds with para-associative ternary products.

Existence of left-invariant vector fields on certain Lie semiheaps.

Generalization of principal bundles to a ternary, heap-like setting.

Abstract

We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known `heapification' functor to the ambience of Lie groups and principal bundles.