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Affine circle geometry over quaternion skew fields. (English) Zbl 0902.51001

The author continues his work from [Abh. Math. Semin. Univ. Hamburg 64, 279-292 (1994; Zbl 0816.51006), Geom. Dedicata 49, No. 2, 239-251 (1994; Zbl 0796.51005), Aequationes Math. 45, No. 2-3, 232-238 (1993; Zbl 0787.51006)].
Let \(L\) be a quaternion skew field and \(K\) a maximal commutative subfield of \(L\), and \(\Sigma(K,L)\) the corresponding chain geometry. As point set of this geometry the author takes rather a spread of the 3-dimensional projective space \({\mathcal P}_K\) over \(K\) than the projective line over \(L\). Taking this point of view the author describes how, by deleting one point \(\infty\) (= spread line of a previously chosen projective subplane \(\widetilde {\mathcal A})\), an affine plane \({\mathcal A}\) arises whose line structure is induced by the chains. A chain \({\mathcal C}\) of \(\Sigma (K,L)\) containing \(\infty\) induces either a line or a degenerated circle of this affine plane according to whether or not \({\mathcal C}\) has a transversal line in \(\widetilde {\mathcal A}\). If \(\infty \notin {\mathcal C}\), a non-degenerate circle arises. Degenerate circles turn out to be affine Baer subplanes of \({\mathcal A}\) coordinatized by the center of \(L\). Non-degenerate circles are intersections of affine Hermitian varieties of \(K\) is a Galois extension of the center of \(L\).

MSC:

51B05 General theory of nonlinear incidence geometry
51B10 Möbius geometries
12F10 Separable extensions, Galois theory
12F99 Field extensions
51A40 Translation planes and spreads in linear incidence geometry
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[1] Benz, W., Vorlesungen über Geometrie der Algebren, (Grundlehren, Bd. 197 (1973), Springer: Springer Berlin) · Zbl 0258.50024
[2] Berger, M., (Geometry 1, II (1987), Springer: Springer Berlin)
[3] Brauner, H., Eine geometrische Kennzeichnung linearer Abbildungen, Monatsh. Math., 77, 10-20 (1973) · Zbl 0256.50013
[4] Cohn, P. M., Quadratic Extensions of Skew Fields, (Proc. London Math. Soc., 11 (1961)), 531-556 · Zbl 0104.03301
[5] Havlicek, H., Die Projektivita̋ten und Antiprojektivitäten der Quaternionengeraden, Publ. Math. Debrecen, 40, 219-227 (1992)
[6] Havlicek, H., On the geometry of field extensions, Aequat. Math., 45, 232-238 (1993) · Zbl 0787.51006
[7] Havlicek, H., Spreads of right quadratic skew field extensions, Geom. Dedicata, 49, 239-251 (1994) · Zbl 0796.51005
[8] Havlicek, H., Spheres of quadratic field extensions, Abh. Math. Sem. Univ. Hamburg, 64, 279-292 (1994) · Zbl 0816.51006
[9] Herzer, A., Chain geometries, (Buekenhout, F., Handbook of Incidence Geometry (1995), Elsevier: Elsevier Amsterdam), 781-842 · Zbl 0829.51003
[10] Herzer, A., Der äquiforme Raum einer Algebra, Mitt. Math. Ges. Hamburg, 13, 129-154 (1993) · Zbl 0798.51009
[11] Kist, G.; Reinmiedl, B., Geradenmodelle der Möbius- und Burau-Geometrien, J. Geometry, 41, 94-113 (1991) · Zbl 0737.51012
[12] Mäurer, H.; Metz, R.; Nolte, W., Die Automorphismengruppe der Möbiusgeometrie einer Körpererweiterung, Aequat. Math., 21, 110-112 (1980) · Zbl 0444.51006
[13] Pickert, G., (Projektive Ebenen (1975), Springer: Springer Berlin)
[14] Schröder, E. M., Metric geometry, (Buekenhout, F., Handbook of Incidence Geometry (1995), Elsevier: Elsevier Amsterdam), 945-1013 · Zbl 0826.51008
[15] Tallini Scafati, M., Metrica hermitiana ellitica in uno spazio proiettivo quaternionale, Ann. Mat. Pura Appl., 60, IV, 203-234 (1963) · Zbl 0114.13001
[16] Wilker, J. B., The Quaternion formalism for Möbius Groups in Four or Fewer Dimensions, Linear Algebra Appl., 190, 99-136 (1993) · Zbl 0786.51005
[17] Wunderlich, W., Ebene Kinematik, (BI Hochschultaschenbücher, 447/447a (1970), Bibliographisches Institut: Bibliographisches Institut Mannheim, Wien, Zürich) · Zbl 0084.17603
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