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.