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Imaginary geometries

Paulien Jansen and Hendrik Van Maldeghem (UGent)
(2023) MODERN MATHEMATICAL METHODS. 1(1). p.43-92
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Abstract
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.
Keywords
Long root geometry, root groups, spherical buildings, Moufang buildings

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MLA
Jansen, Paulien, and Hendrik Van Maldeghem. “Imaginary Geometries.” MODERN MATHEMATICAL METHODS, vol. 1, no. 1, 2023, pp. 43–92.
APA
Jansen, P., & Van Maldeghem, H. (2023). Imaginary geometries. MODERN MATHEMATICAL METHODS, 1(1), 43–92.
Chicago author-date
Jansen, Paulien, and Hendrik Van Maldeghem. 2023. “Imaginary Geometries.” MODERN MATHEMATICAL METHODS 1 (1): 43–92.
Chicago author-date (all authors)
Jansen, Paulien, and Hendrik Van Maldeghem. 2023. “Imaginary Geometries.” MODERN MATHEMATICAL METHODS 1 (1): 43–92.
Vancouver
1.
Jansen P, Van Maldeghem H. Imaginary geometries. MODERN MATHEMATICAL METHODS. 2023;1(1):43–92.
IEEE
[1]
P. Jansen and H. Van Maldeghem, “Imaginary geometries,” MODERN MATHEMATICAL METHODS, vol. 1, no. 1, pp. 43–92, 2023.
@article{01HYYY408M3XQBK1NKN8J8GN2Y,
  abstract     = {{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.}},
  author       = {{Jansen, Paulien and Van Maldeghem, Hendrik}},
  issn         = {{3023-5294}},
  journal      = {{MODERN MATHEMATICAL METHODS}},
  keywords     = {{Long root geometry,root groups,spherical buildings,Moufang buildings}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{43--92}},
  title        = {{Imaginary geometries}},
  url          = {{https://modernmathmeth.com/index.php/pub/article/view/11/5}},
  volume       = {{1}},
  year         = {{2023}},
}