Split Cayley hexagons via subalgebras of octonion algebras
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Norbert Knarr
Abstract
We give a description of the split Cayley hexagon, using the split Cayley algebra of octonions, and its subalgebras of dimension five or six as points or lines, respectively. That description is then used to identify various interesting subgeometries, and a local isomorphism from the hexagon onto a projective plane.
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Communicated by: H. Cuypers
References
[1] A. Blunck, N. Knarr, B. Stroppel, M. J. Stroppel, Transitive groups of similitudes generated by octonions. J. Group Theory 21 (2018), 1001–1050. MR3871471 Zbl 1439.2005910.1515/jgth-2018-0018Search in Google Scholar
[2] A. Blunck, S. Pasotti, S. Pianta, Generalized Clifford parallelisms. Innov. Incidence Geom. 11 (2010), 197–212. MR2795063 Zbl 1260.5100110.2140/iig.2010.11.197Search in Google Scholar
[3] R. Gramlich, H. Van Maldeghem, Epimorphisms of generalized polygons. II. Some existence and non-existence results. In: Finite geometries, volume 3 of Dev. Math., 177–200, Kluwer 2001. MR2061805 Zbl 1017.5100810.1007/978-1-4613-0283-4_12Search in Google Scholar
[4] T. Grundhöfer, M. J. Stroppel, H. Van Maldeghem, Embeddings of hermitian unitals into pappian projective planes. Aequationes Math. 93 (2019), 927–953. MR4008656 Zbl 1430.5100610.1007/s00010-019-00652-xSearch in Google Scholar
[5] T. Grundhöfer, M. J. Stroppel, H. Van Maldeghem, Linear spaces embedded into projective spaces via Baer sublines. Comb. Theory 2 (2022), Paper No. 4, 14 pages. MR4405993 Zbl 1507.5100210.5070/C62156876Search in Google Scholar
[6] N. Knarr, Projectivities of generalized polygons. Ars Combin. 25 (1988), 265–275. MR942482 Zbl 0654.51016Search in Google Scholar
[7] N. Knarr, M. J. Stroppel, Subalgebras of octonion algebras. J. Algebra 664 (2025), 42–74. MR4812310 Zbl 0797618110.1016/j.jalgebra.2024.10.004Search in Google Scholar
[8] M. A. Ronan, A geometric characterization of Moufang hexagons. Invent. Math. 57 (1980), 227–262. MR568935 Zbl 0429.5100210.1007/BF01418928Search in Google Scholar
[9] R. D. Schafer, An introduction to nonassociative algebras. Academic Press 1966. MR210757 Zbl 0145.25601Search in Google Scholar
[10] G. J. Schellekens, On a hexagonic structure. I. Indag. Math. 24 (1962), 201–217. MR143075 Zbl 0105.1300110.1016/S1385-7258(62)50019-XSearch in Google Scholar
[11] T. A. Springer, F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups. Springer 2000. MR1763974 Zbl 1087.1700110.1007/978-3-662-12622-6Search in Google Scholar
[12] J. Tits, Sur la trialité et certains groupes qui s’en déduisent. Inst. Hautes Études Sci. Publ. Math. no. 2 (1959), 13–60. MR1557095 Zbl 0088.3720410.1007/BF02684706Search in Google Scholar
[13] F. van der Blij, T. A. Springer, Octaves and triality. Nieuw Arch. Wisk. (3) 8 (1960), 158–169. MR123622 Zbl 0127.11804Search in Google Scholar
[14] H. Van Maldeghem, Generalized polygons. Springer 1998. MR3014920 Zbl 1235.5100210.1007/978-3-0348-0271-0Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
