Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-10-31T23:29:09.214Z Has data issue: false hasContentIssue false
  • English
  • Français

On exceptional Lie geometries

Part of: Nonlinear incidence geometry Finite geometry and special incidence structures Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  11 January 2021

Anneleen De Schepper ,
Jeroen Schillewaert ,
Hendrik Van Maldeghem  and
Magali Victoor
Anneleen De Schepper
Affiliation:
Department of Mathematics, Ghent University, 9000GhentBelgium; E-mail: anneleen.deschepper@ugent.be.
Jeroen Schillewaert
Affiliation:
Department of Mathematics, University of Auckland, 1010AucklandNew Zealand; E-mail: j.schillewaert@auckland.ac.nz.
Hendrik Van Maldeghem
Affiliation:
Department of Mathematics, Ghent University, 9000GhentBelgium; E-mail: hendrik.vanmaldeghem@ugent.be.
Magali Victoor
Affiliation:
Department of Mathematics, Ghent University, 9000GhentBelgium; E-mail: magali.victoor@ugent.be.

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.

MSC classification

Primary: 51E24: Buildings and the geometry of diagrams
Secondary: 51B25: Lie geometries 20E42: Groups with a $BN$-pair; buildings
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e2
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Brouwer, A. E. and Cohen, A. M., Some remarks on Tits geometries, Indag. Math. 45 (1983), 393–400.CrossRefGoogle Scholar
Bourbaki, N., Groupes et Algèbres de Lie, Chapters 4, 5 and 6, Actu. Sci. Ind. 1337, Hermann, Paris (1968).Google Scholar
Buekenhout, F. and Cohen, A., Diagram geometries related to classical groups and buildings, EA Series of Modern Surveys in Mathematics 57. Springer, Heidelberg, (2013), xiv+592 pp.Google Scholar
Cohen, A. M. and Cooperstein, B., A characterization of some geometries of Lie type, Geom. Dedicata 15 (1983), 73–105.CrossRefGoogle Scholar
Cohen, A. M. and Cooperstein, B., On the local recognition of finite metasymplectic spaces, J. Algebra 124 (1989), 348–366.CrossRefGoogle Scholar
Cohen, A. M., De Schepper, A., Schillewaert, J. and Van Maldeghem, H., Shult’s haircut theorem revised, submitted.Google Scholar
Cooperstein, B., A characterization of some Lie incidence structures, Geom. Dedicata 6 (1977), 205–258.CrossRefGoogle Scholar
De Schepper, A., Schillewaert, J., Van Maldeghem, H. and Victoor, M., ‘A geometric characterisation of Hjelmslev-Moufang planes’, submitted, https://cage.ugent.be/~ads/links/5.pdf.Google Scholar
Freudenthal, H., Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145–190.CrossRefGoogle Scholar
Freudenthal, H., Symplektische und metasymplektische Geometrien, in Algebraical and Topological Foundations of Geometry (Proc. Colloq., Utrecht, 1959) Pergamon, Oxford (1962), 29–33.CrossRefGoogle Scholar
Kasikova, A. and Shult, E. E., Point-line characterizations of Lie geometries, Adv. Geom. 2 (2002), 147–188.CrossRefGoogle Scholar
Krauss, O., Schillewaert, J. and Van Maldeghem, H., Veronesean representations of Moufang planes, Mich. Math. J. 64 (2015), 819–847.CrossRefGoogle Scholar
Schillewaert, J. and Van Maldeghem, H., Projective planes over quadratic two-dimensional algebras, Adv. Math. 262 (2014), 784–822.CrossRefGoogle Scholar
Schillewaert, J. and Van Maldeghem, H., Imbrex geometries, J. Combin. Theory Ser. A. 127 (2014), 286–302.CrossRefGoogle Scholar
Schillewaert, J. and Van Maldeghem, H., On the varieties of the second row of the split Freudenthal-Tits Magic Square, Ann. Inst. Fourier 67 (2017), 2265–2305.CrossRefGoogle Scholar
Shult, E. E., On characterizing the long-root geometries, Adv. Geom. 10 (2010), 353–370.CrossRefGoogle Scholar
Shult, E. E., Characterizing the half-spin geometries by a class of singular subspaces, Bull. Belg. Math. Soc. 12 (2005), 883–894.CrossRefGoogle Scholar
Shult, E. E., Points and Lines, Characterizing the Classical Geometries, Universitext, Springer-Verlag, Berlin Heidelberg (2011), xxii+676 pp.Google Scholar
Shult, E. E., Parapolar spaces with the ‘Haircut’ axiom, Innov. Incid. Geom. 15 (2017), 265–286.CrossRefGoogle Scholar
Tits, J., Groupes semi-simples complexes et géométrie projective, Séminaire Bourbaki 7 (1954/1955).Google Scholar
Tits, J., Sur certaines classes d’espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8 ${}^o$(2) 29 (1955).Google Scholar
Tits, J., Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Indag. Math. 28 (1966), 223–237.CrossRefGoogle Scholar
Tits, J., Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer-Verlag, Berlin-New York (1974), x+299 pp.Google Scholar
Tits, J., A local approach to buildings. in The Geometric Vein. The Coxeter Festschrift, Coxeter Symposium, University of Toronto, 21–25 May 1979, Springer-Verlag, New York 1982, 519–547.CrossRefGoogle Scholar