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Quaternion Hermitian Planes

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The quaternion hermitian planes are defined, and are characterized by certain groups of automorphisms. For this purpose, characterizations of locally compact connected translation planes (in the context of stable planes) and compact connected projective desarguesian planes are given.

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5. References

  1. André, J., ‘Über nicht-Desarguessche Bbenen mit transitiver Translationsgruppe’, Math. Z. 60 (1954) 156–186.

    Article  MathSciNet  MATH  Google Scholar 

  2. Betten, D., ‘4-dimensionale Translationsebenen mit kommutativer Standgruppe’, Math. Z. 154 (1977) 121–141.

    Article  MathSciNet  Google Scholar 

  3. Dugundji, J., Topology, Allyn and Bacon, Boston, 1966.

    MATH  Google Scholar 

  4. Freudenthal, H., ‘Einige Sätze über topologische Gruppen’, Ann. of Math. 37 (1936) 46–56.

    Article  MathSciNet  Google Scholar 

  5. Grundhöfer, T., and Stroppel, M., ‘On restrictions of groups of automorphisms of compact connected projective planes to subplanes’, Res. Math. 21 (1992) 319–327.

    Article  MATH  Google Scholar 

  6. Hähl, H., ‘Achtdimensionale lokalkompakte Translationsebenen mit mindestens 17-dimensionaler Kollineationsgruppe’, Geom. Ded. 21 (1986) 299–340.

    Article  MATH  Google Scholar 

  7. Hähl, H., ‘Die Oktavenebene als Translationsebene mit gro\er Kollineationsgruppe’, Monatsh. Math. 106 (1988) 265–299.

    Article  MathSciNet  MATH  Google Scholar 

  8. Halder, H.R., ‘Die Dimension der Bahnen lokalkompakter Gruppen’, Arch, der Math. 22 (1971) 302–303.

    Article  MathSciNet  MATH  Google Scholar 

  9. Husain, T., Introduction to topological groups, Saunders, Philadelphia, London, 1966.

    MATH  Google Scholar 

  10. Knarr, N., ‘4-dimensionale projektive Ebenen mit gro\er abelscher Kollineationsgruppe’, J. Geom. 31 (1988) 114–124.

    Article  MathSciNet  MATH  Google Scholar 

  11. Löwen, R., ‘Vierdimensionale stabile Ebenen’, Geom. Ded. 5 (1976) 239–294.

    MATH  Google Scholar 

  12. Löwen, R., ‘Halbeinfache Automorphismengruppen von vierdimensionalen stabilen Ebenen sind quasi-einfach’, Math. Ann. 236 (1978) 15–28.

    Article  MathSciNet  MATH  Google Scholar 

  13. Löwen, R., ‘Central collineations and the Parallel Axiom in stable planes’, Geom. Ded. 10 (1981) 283–315.

    Article  MATH  Google Scholar 

  14. Löwen, R., ‘Stable planes of low dimension admitting reflections at many lines’, Res. Math. 5 (1982) 60–80.

    Article  MATH  Google Scholar 

  15. Löwen, R., ‘Topology and dimension of stable planes: On a conjecture by H. Freudenthal’, J. Reine Angew. Math. 343 (1983) 108–122.

    MathSciNet  MATH  Google Scholar 

  16. Löwen, R., ‘Actions of Spin3 on 4-dimensional stable planes’, Geom. Ded. 21 (1986) 1–12.

    Article  MATH  Google Scholar 

  17. Löwen, R., ‘Stable Planes admitting a Classical Motion Group’, Res. Math. 9 (1986) 119–130.

    Article  MATH  Google Scholar 

  18. Lüneburg, H., Translation planes, Springer, Berlin etc., 1980.

    Book  MATH  Google Scholar 

  19. Mann, L.N., ‘Gaps in the dimensions of compact transformation groups’, Illinois J. Math. 10 (1966) 532–546.

    MathSciNet  MATH  Google Scholar 

  20. Pall, G., ‘Hermitian quadratic forms in a quasi-field’, Bull. Am. Math. Soc. 51 (1945) 889–893.

    Article  MathSciNet  MATH  Google Scholar 

  21. Pickert, G., Projektive Ebenen, Springer, Berlin etc., 1955.

    Book  MATH  Google Scholar 

  22. Pontrjagin, L.S., ‘Über stetige algebraische Körper’, Ann. Math. 33 (1932) 163–174.

    Article  MathSciNet  Google Scholar 

  23. Salzmann, H., ‘Topological Planes’, Adv. in Math. 2 (1969) 1–60.

    Article  MathSciNet  Google Scholar 

  24. Salzmann, H., ‘Compact 8-dimensional projective planes with large collineation groups’, Geom. Ded. 8 (1979) 139–161.

    Article  MathSciNet  MATH  Google Scholar 

  25. Skornjakov, L.A., ‘Topological projective planes’, Trudy Moskov. Mat. Obšč. 3 (1954) 347–373.

    Google Scholar 

  26. Stroppel, M., Achtdimensionale stabile Ebenen mit quasieinfacher Automorphismengruppe, Dissertation, Tübingen, 1991.

  27. Stroppel, M., ‘Reconstruction of incidence geometries from groups of automorphisms’, Archiv der Math. (1992) (to appear).

  28. Stroppel, M., ‘Planar groups of automorphisms of stable planes’, J. Geom. (1992) (to appear).

  29. Stroppel, M., Quasi-perspectivities in stable planes, Preprint No. 1462, Math. Institut der Technischen Hochschule Darmstadt, 1992.

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Correspondence to Markus Stroppel.

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Stroppel, M. Quaternion Hermitian Planes. Results. Math. 23, 387–397 (1993). https://doi.org/10.1007/BF03322312

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