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A geometric characterization of the Hjelmslev-Moufang planes

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Abstract
Hjelmslev-Moufang planes are point-line geometries related to the exceptional algebraic groups of type E6. 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 Hjelmslev-Moufang 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 Hjelmslev-Moufang planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to these structures with Jordan algebras due to Faulkner in his book “The Role of Nonassociative Algebra in Projective Geometry”, published by the AMS in 2014; on the other hand, it provides a new tool for classification and characterisation problems in the general theory of parapolar spaces
Keywords
Hjelmslev-Moufang planes, Lie incidence geometries, parapolar spaces, E6

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MLA
De Schepper, Anneleen, et al. “A Geometric Characterization of the Hjelmslev-Moufang Planes.” QUARTERLY JOURNAL OF MATHEMATICS, vol. 73, no. 1, 2022, pp. 369–94, doi:10.1093/qmath/haab043.
APA
De Schepper, A., Schillewaert, J., Van Maldeghem, H., & Victoor, M. (2022). A geometric characterization of the Hjelmslev-Moufang planes. QUARTERLY JOURNAL OF MATHEMATICS, 73(1), 369–394. https://doi.org/10.1093/qmath/haab043
Chicago author-date
De Schepper, Anneleen, Jeroen Schillewaert, Hendrik Van Maldeghem, and Magali Victoor. 2022. “A Geometric Characterization of the Hjelmslev-Moufang Planes.” QUARTERLY JOURNAL OF MATHEMATICS 73 (1): 369–94. https://doi.org/10.1093/qmath/haab043.
Chicago author-date (all authors)
De Schepper, Anneleen, Jeroen Schillewaert, Hendrik Van Maldeghem, and Magali Victoor. 2022. “A Geometric Characterization of the Hjelmslev-Moufang Planes.” QUARTERLY JOURNAL OF MATHEMATICS 73 (1): 369–394. doi:10.1093/qmath/haab043.
Vancouver
1.
De Schepper A, Schillewaert J, Van Maldeghem H, Victoor M. A geometric characterization of the Hjelmslev-Moufang planes. QUARTERLY JOURNAL OF MATHEMATICS. 2022;73(1):369–94.
IEEE
[1]
A. De Schepper, J. Schillewaert, H. Van Maldeghem, and M. Victoor, “A geometric characterization of the Hjelmslev-Moufang planes,” QUARTERLY JOURNAL OF MATHEMATICS, vol. 73, no. 1, pp. 369–394, 2022.
@article{8745694,
  abstract     = {{Hjelmslev-Moufang planes are point-line geometries related to the exceptional algebraic groups of type E6. 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 Hjelmslev-Moufang 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 Hjelmslev-Moufang planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to these structures with Jordan algebras due to Faulkner in his book “The Role of Nonassociative Algebra in Projective Geometry”, published by the AMS in 2014; on the other hand, it provides a new tool for classification and characterisation problems in the general theory of parapolar spaces}},
  author       = {{De Schepper, Anneleen and Schillewaert, Jeroen and Van Maldeghem, Hendrik and Victoor, Magali}},
  issn         = {{0033-5606}},
  journal      = {{QUARTERLY JOURNAL OF MATHEMATICS}},
  keywords     = {{Hjelmslev-Moufang planes,Lie incidence geometries,parapolar spaces,E6}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{369--394}},
  title        = {{A geometric characterization of the Hjelmslev-Moufang planes}},
  url          = {{http://doi.org/10.1093/qmath/haab043}},
  volume       = {{73}},
  year         = {{2022}},
}

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