Polarities and planar collineations of Moufang planes. (English) Zbl 1281.51003
The authors study collineations and correlations of Moufang planes. They show that conjugacy classes of Baer involutions are not refined if one restricts the conjugating element to the stabilizer of a quadrangle, and then they establish an interrelation between Baer involutions and polarities. Finally, they use their results to prove a transitivity result for centralizers of polarities (Theorem 5.3).
Reviewer: Vito Napolitano (Napoli)
MSC:
| 51A35 | Non-Desarguesian affine and projective planes |
| 51A10 | Homomorphism, automorphism and dualities in linear incidence geometry |
| 17A35 | Nonassociative division algebras |
| 17A36 | Automorphisms, derivations, other operators (nonassociative rings and algebras) |
| 17A75 | Composition algebras |
| 51A40 | Translation planes and spreads in linear incidence geometry |
Keywords:
Moufang plane; translation plane; Baer involution; polarity; conjugacy; semifield; division algebra; alternative algebra; composition algebra; octonion field; automorphism; autotopismReferences:
| [1] | Baer R.: Polarities in finite projective planes. Bull. Am. Math. Soc. 52, 77–93 (1946). doi: 10.1090/S0002-9904-1946-08506-7 · Zbl 0060.32308 · doi:10.1090/S0002-9904-1946-08506-7 |
| [2] | Bruck R.H., Kleinfeld E.: The structure of alternative division rings. Proc. Am. Math. Soc. 2, 878–890 (1951). doi: 10.2307/2031702 · Zbl 0044.02205 · doi:10.1090/S0002-9939-1951-0045099-9 |
| [3] | Gulde, M., Stroppel, M.: Stabilizers of subspaces under similitudes of the Klein quadric, and automorphisms of Heisenberg algebras. Linear Algebra Appl. (2012). doi: 10.1016/j.laa.2012.03.018 . arXiv:1012.0502 · Zbl 1300.17009 |
| [4] | Hähl H.: Sechzehndimensionale lokalkompakte Translationsebenen mit Spin(7) als Kollineationsgruppe. Arch. Math. (Basel) 48(3), 267–276 (1987). doi: 10.1007/BF01195360 · Zbl 0639.51017 · doi:10.1007/BF01195360 |
| [5] | Hughes, D.R., Piper, F.C.: Projective planes. In: Graduate Texts in Mathematics, vol. 6. Springer, New York (1973) · Zbl 0267.50018 |
| [6] | Kleinfeld, E.: Alternative division rings of characteristic 2. Proc. Nat. Acad. Sci. USA 37, 818–820 (1951). http://www.pnas.org/content/37/12/818.full.pdf · Zbl 0044.02301 |
| [7] | Knarr, N., Stroppel, M.: Baer involutions and polarities in Moufang planes of characteristic two. Adv. Geom. (2012, to appear) · Zbl 1283.51002 |
| [8] | Lang S.: Algebra, 2 edn. Addison-Wesley Publishing Company Advanced Book Program, Reading (1984) |
| [9] | Mäurer H., Stroppel M.: Groups that are almost homogeneous. Geom. Dedicata 68(2), 229–243 (1997). doi: 10.1023/A:1005090519480 · Zbl 0890.20025 · doi:10.1023/A:1005090519480 |
| [10] | Pickert G.: Projektive Ebenen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. LXXX. Springer, Berlin (1955) |
| [11] | Salzmann, H.: Viereckstransitivität der kleinen projektiven Gruppe einer Moufang-Ebene. Ill. J. Math. 3, 174–181 (1959). http://projecteuclid.org/getRecord?id=euclid.ijm/1255455119 · Zbl 0085.14203 |
| [12] | Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R., Stroppel, M.: Compact projective planes. In: de Gruyter Expositions in Mathematics, vol. 21. Walter de Gruyter & Co., Berlin (1995). With an introduction to octonion geometry · Zbl 0851.51003 |
| [13] | Springer, T.A., Veldkamp, F.D.: Octonions, Jordan algebras and exceptional groups. In: Springer Monographs in Mathematics. Springer, Berlin (2000) · Zbl 1087.17001 |
| [14] | Stroppel M.: Homogeneous symplectic maps and almost homogeneous Heisenberg groups. Forum Math. 11(6), 659–672 (1999). doi: 10.1515/form.1999.018 · Zbl 0928.22008 · doi:10.1515/form.1999.018 |
| [15] | Stroppel M.: Locally compact groups with many automorphisms. J. Group Theory 4(4), 427–455 (2001). doi: 10.1515/jgth.2001.032 · Zbl 0999.22006 · doi:10.1515/jgth.2001.032 |
| [16] | Tits, J., Weiss, R.M.: Moufang polygons. In: Springer Monographs in Mathematics. Springer, Berlin (2002) · Zbl 1010.20017 |
| [17] | Völklein H.: Fahnentransitive lineare Gruppen mit diagonalisierbarer Standgruppe. Geom. Dedicata 13(2), 215–229 (1982). doi: 10.1007/BF00147664 · Zbl 0507.20019 · doi:10.1007/BF00147664 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
