Affine Reductive Spaces of Small Dimension and Left A-Loops

Abstract.

In this paper we determine the at least 4-dimensional affine reductive homogeneous manifolds for an at most 9-dimensional simple Lie group or an at most 6-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable left A-loops. Using this we classify all global almost differentiable left A-loops L having either a 6-dimensional semi-simple Lie group or the group ${SL}_3(\mathbb{R})$ as the group topologically generated by their left translations. Moreover, we determine all at most 5-dimensional left A-loops L with ${PSU}_3(\mathbb{C},1)$ as the group topologically generated by their left translations.