Cross-ratios in Moufang--Klingenberg planes
The author generalizes some well known connections between cross-ratio and geometric transformation groups to the case of cohesive Moufang- Klingenberg planes. Such a plane \(\mathcal M\) can be represented over a local alternative chain ring \(\mathcal R\) and then the projective line \(\mathbb{P}(\mathcal R)\) can be considered as a line \(g\) of \(\mathcal M\). If the conjugacy relation in \(\mathcal R\) is transitive, then in a standard way we define the cross-ratio on \(g\). The author investigates several transformation groups of \(g\): \(\Lambda\) generated by linear transformations and inverses, group of projectivities in \(\Pi\), \(\Gamma\) determined by collineations of \(\mathcal M\) fixing \(g\), \(\Delta\) determined by collineations and dualities, and \(\Phi\) consisting of bijections preserving cross-ratio. Clearly \(\Lambda\leq \Pi\leq \Gamma\) and \(\Gamma\) is a normal subgroup \(\Delta\). Under certain assumptions on \(\mathcal R\) the author proves the following deeper results of this kind. If \({\mathcal U}={\mathcal R}/I\neq \text{GF}(2)\), where \(I\) is the ideal of non-units of \(\mathcal R\), then \(\Lambda=\Pi\). The groups \(\Delta\) and \(\Gamma\) are determined by \(\Pi\) and special Jordan automorphisms of \(\mathcal R\). If a ring of dual numbers \(\mathcal R=\mathcal U(\varepsilon)\) \((\varepsilon^ 2=0)\) is not associative, then \(\Phi\leq \Delta\); more precisely it is determined by \(\Pi\) and standard anti- automorphism of \(\mathcal R\).
scientific article; zbMATH DE number 64291