Bol loops as sections in semi-simple Lie groups of small dimension

TL;DR

This paper classifies small-dimensional Bol loops within semi-simple Lie groups, revealing their structure as products or extensions of hyperbolic Bruck loops, thus advancing understanding of their geometric and algebraic properties.

Contribution

It provides a complete classification of connected differentiable Bol loops with semi-simple Lie groups of dimension up to 9, linking them to known loop structures.

Findings

All such Bol loops are isotopic to products of hyperbolic Bruck loops.

They are also characterized as Scheerer extensions of Lie groups by these loops.

The classification connects Bol loops with affine symmetric spaces.

Abstract

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most $9$-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.