A Geometric Characterization of the Hjelmslev–Moufang Planes

Abstract

Abstract Hjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $\mathsf{E}_6$. More generally, point-line geometries related to spherical Tits buildings—Lie incidence geometries—are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar behaviour as the HM planes, in the sense that their symplecta never have a non-empty intersection. Under standard assumptions, we obtain that the only such parapolar spaces are exactly given by the HM planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to HM planes using Jordan algebras and due to Faulkner in his book ‘The Role of Nonassociative Algebra in Projective Geometry’, published by the American Mathematical Society in 2014; on the other hand, it provides a new tool for classification and characterization problems in the general theory of parapolar spaces.