Affine Extensions of loops

Abstract

We show a simple geometric procedure for an extension of a loop realized as the image ${\Sigma }^{\ast }$ of a sharply transitive section in a subgroup $G^{\ast }$ of the projective linear group $PGL(n-1,\mathbb{K})$ to a loop realized as the image of a sharply transitive section in a group $\Delta =T^{\prime }⋊C$ of affinities of the $n$-dimensional space ${\mathcal{A}}_n={\mathbb{K}}^n$ over a commutative field $\mathbb{K}$. We desire that $T^{\prime }$ is a large subgroup of affine translations and that $\alpha (C)=G^{\ast }$ holds for the canonical homomorphism $\alpha :GL(n,\mathbb{K})\to PGL(n,\mathbb{K})$. We demonstrate that our construction successfully can be applied to sharply transitive sections in unitary and orthogonal groups $S{U}_{p_2}(n,F)$ of positive index $p_2$ over ordered pythagorean $n$-real fields $F$.