Affine Extensions of loops

TL;DR

This paper presents a geometric method to extend loops realized in projective linear groups to affine groups, enabling new constructions in affine and unitary/orthogonal groups over specific fields.

Contribution

It introduces a simple geometric procedure for extending loops from projective linear groups to affine groups, broadening the scope of loop realizations in affine and classical groups.

Findings

Successful application to unitary groups $SU_{p_2}(n,F)$

Extension method preserves sharp transitivity in affine groups

Applicable over ordered Pythagorean $n$-real fields

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 $\Delta =T' \rtimes C$ of affinities of the $n$-dimensional space ${\cal A}_n=\mathbb K^n$ over a commutative field $\mathbb K$. We desire that $T'$ 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 $SU_{p_2}(n,F)$ of positive index $p_2$ over ordered pythagorean $n$-real fields $F$.