A Geometric Characterization of the Hjelmslev–Moufang Planes

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.

On exceptional Lie geometries

“…Our main results are as follows. We combine them with the results from [7] in order to also include the (−1)-lacunary case. For a more detailed version, mentioning the index k for which the parapolar space is k-lacunary, we refer to Tables 1 and 2.…”

“…By Lemma 6.1, the point-residual Ω p , for any p ∈ X , is a strong parapolar space with lacunary index −1. By the main result of [7], see also…”

“…Since any two symps through p have rank at least 3 and share a line, there is only one such component. Then by column k = −1 of Table 1, proved in [7], every line through p is contained in at least two symps. Hence every point of Ω collinear to p is contained in at least two symps and we can interchange its role with that of p. A connectivity argument now shows that Ω x is a strong parapolar space with lacunary index −1, for every x ∈ X .…”

“…Such (strong) parapolar spaces are classified in [7]. That result, together with the classification of 0lacunary strong parapolar spaces admitting symplecta of rank 2, is then taken as the first step of an inductive process to determine all lacunary parapolar spaces, see Subsection 1.5 below.…”

“…From then on, we proceed inductively. Section 6 deals with the 0-lacunary case, and partially builds on the main result of [7], namely, when all symps have rank at least 3. If some symps have rank 2, then we show that either we have a so-called imbrex geometry and we can use [13], or the diameter of the parapolar space is 3.…”