n-spaces

Year 2020, Volume: 69 Issue: 1, 431 - 440, 30.06.2020

Atilla Akpınar

https://doi.org/10.31801/cfsuasmas.668538

Abstract

In this paper, we introduce n-spaces constructed over an local ring with the maximal ideal (of non-unit elements). So, we give the example of an octonion n-space. Finally, we give two collineations of quaternion n-space.

Keywords

Local ring, projective Klingenberg plane, n-space

References

• Akpinar, A. and Erdogan, F.O., Dual Quaternionic (n-1)-Spaces Defined by Special Jordan Algebras of Dimension 4n²-2n, JP Journal of Geometry and Topology, 21(4) (2018), 327-364.
• Baker, C.A., Lane N.D. and Lorimer, J.W., A coordinatization for Moufang-Klingenberg Planes, Simon Stevin, 65 (1991), 3-22.
• Bix, R., Octonion Planes over Local Rings, Trans. Amer. Math. Soc., 261(2) (1980), 417-438.
• Blunck, A., Cross-ratios in Moufang-Klingenberg Planes, Geom. Dedicata, 43 (1992), 93-107.
• Celik, B., Akpinar, A. and Ciftci, S., 4-Transitivity and 6-figures in some Moufang-Klingenberg planes, Monatshefte für Mathematik, 152(4) (2007), 283-294.
• Faulkner, J.R., Octonion Planes Defined by Quadratic Jordan Algebras, Mem. Amer. Math. Soc., 104 (1970), 1-71.
• Faulkner, J.R., The Role of Nonassociative Algebra in Projective Geometry, Graduate Studies in Mathematics, Vol. 159, Amer. Math. Soc., Providence, R.I., 2014.
• Freudenthal, H., Octaven, Ausnahmegruppen, und Octavengeometrie. Mathematisch Instituut der Rijksuniversiteit te Utrecht, Utrecht, 49, 1951.
• Jacobson, N., Structure and Representations of Jordan Algebras, Colloq. Publ. 39, Amer. Math. Soc., Providence, R.I., 1968.
• Jordan, P., Über Eine Nicht-Desarguessche Ebene Projektive Geometrie. Abh. Math. Sem. Univ. Hamburg, 16 (1949), 74-76, . McCr : McCrimmon, K., The Freudenthal-Springer-Tits Constructions of Exceptional Jordan Algebras, Trans. of the Amer. Math. Soc., 139 (1969), 495-510, .
• Springer, T.A., The Projective Octave Plane. Nederl. Akad. Wetensch. Proc. Ser. A, 63, Indag. Math., 22 (1960), 74-101.