Imaginary geometries



Long root geometry, root groups, spherical buildings, Moufang buildings


In this paper, we axiomatize the geometries obtained from the long root subgroup geometries by taking as new lines the so-called imaginary lines. A generic such line is the union of the orbits of the centers of the two root groups corresponding to two opposite long roots, which share at least two points. This extends characterizations of Cuypers and Hall on copolar spaces, who treated the quadrangular case. Here, we treat the remaining case, the hexagonal one. Our results hold over any field of size at least 5 and characteristic different from 2.


P. Abramenko, K. Brown: Buildings: Theory and Applications, Graduate Texts in Mathematics 248, Springer (2008).

R. J. Blok: Highest weight modules and polarized embeddings of shadow spaces, J. Alg. Combin., 34 (2011), 67–113.

A. M. Cohen, G. Ivanyos: Root filtration spaces from Lie algebras and abstract root groups, J. Algebra, 300 (2006), 433–454.

A. M. Cohen, G. Ivanyos: Root shadow spaces, European J. Combin., 28 (2007), 1419–1441.

A. M. Cohen, A. Steinbach, R. Ushirobira and D. Wales: Lie algebras generated by extremal elements, J. Algebra, 236 (2001), 122–154.

B. N. Cooperstein: A characterization of some Lie incidence structures, Geom. Dedicata, 6 (1977), 205–258.

H. Cuypers: The geometry of k-transvection groups, J. Algebra, 300 (2006), 455–471.

H. Cuypers: The geometry of hyperbolic lines in polar spaces, (2018), arXiv:1707.02099.

H. Cuypers, J. Meulewaeter: Extremal elements in Lie algebras, buildings and structurable algebras, J. Algebra, 580 (2021), 1–42.

A. De Schepper, J. Schillewaert, H. Van Maldeghem and M. Victoor: On exceptional Lie geometries, Forum Math. Sigma, 9 (2) (2021), 1–27.

J. I. Hall: Classifying copolar spaces and graphs, Quart. J. Math., 33 (1982), 421–449.

P. Jansen: Abelian Tits sets of index type, Ph.D. Thesis, Ghent University, (2023).

P. Jansen, H. Van Maldeghem: Parapolar spaces of infinite rank, to appear in Adv. Stud. Euro-Tbil. Math. J..

P. Jansen, H. Van Maldeghem: Subgeometries of (exceptional) Lie incidence geometries induced by maximal root subsystems, preprint #274 at

B. Mühlherr, R. M. Weiss: Receding polar regions of a spherical building and the center conjecture, Ann. Inst. Fourier, 63 (2013), 479-513.

B. Mühlherr, R. M. Weiss: Freudenthal triple systems in arbitrary characteristic, J. Algebra, 520 (2019), 237–275.

B. Mühlherr, R. M. Weiss: Root graded groups of rank 2, J. Comb. Algebra, 3 (2019), 189–214.

B. Mühlherr, R. M. Weiss: Tits polygons, Mem. Amer. Math. Soc., 275 (1352), Providence, (2022).

B. Mühlherr, R. M. Weiss: The exceptional Tits quadrangles revisited, Transform. Groups, (2022),

M. A. Ronan: A geometric characterization of Moufang hexagons, Invent. Math., 57 (1980), 227–262.

M. A. Ronan: A combinatorial characterization of the dual Moufang hexagons, Geom. Dedicata, 11 (1981), 61–67.

M. A. Ronan: Lecture notes on buildings, University of Chicago Press, Chicago, IL (2009).

E. E. Shult: On characterizing the long-root geometries, Adv. Geom., 10 (2010), 353–370.

E. E. Shult: Points and Lines: Characterizing the Classical Geometries, Universitext, Springer-Verlag, Berlin Heidelberg (2011).

F. G. Timmesfeld: Abstract root subgroups and Simple Groups of Lie-Type, Monographs in Mathematics 95, Birkhäuser Basel (2001).

J. Tits: Buildings of Spherical Type and Finite BN-Pairs, Springer Lecture Notes Series 386, Springer-Verlag (1974).

J. Tits: Groupes de rang 1 et ensembles de Moufang, Résumè de cours, Annuaire du Collège de France, 100 (1999-2000), 93–109.

H. Van Maldeghem: Generalized Polygons, Modern Birkhäuser Classics, Birkhäuser (1998).

J. Tits, R. Weiss: Moufang Polygons, Springer Monographs in Mathematics, Springer (2002).




How to Cite

Jansen, P., & Van Maldeghem, H. (2023). Imaginary geometries. Modern Mathematical Methods, 1(1), 43–92. Retrieved from