Two-sided fundamental theorem of affine geometry (Q2101704)

The main result of this paper is the following version of the fundamental theorem of affine geometry for two-sided affine subspaces. Let \(k\in \{2, \ldots, n-1\}\), and let \(f : K^n\to K^n\), with \(n\geq 3\), be a bijective map that takes every left or right affine subspace of dimension \(k\) to a left or right affine subspace of dimension \(\leq k\). The image of a left affine subspace may be right and vice versa. Then \(f\) is the composition of multiplication on the right by an \(a\in K \setminus \{0\}\), an isomorphism of \(K\)-bimodules, a translation, and an automorphism or anti-automorphism of \(K\) applied component-wise. In particular, if \(K = \mathbb{H}\), the skew field of quaternions, then there exist an automorphism \(g : \mathbb{H}^n\to \mathbb{H}^n\) of \(\mathbb{H}\)-bimodules, and elements \(a, q\in \mathbb{H} \setminus \{0\}\) and \(b\in \mathbb{H}^n\) such that \(f\) is the map \[ x\mapsto q(g(xa) + b)q^{-1}, \] possibly composed with quaternion conjugation.