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Characterising Clifford parallelisms among Clifford-like parallelisms. (English) Zbl 1464.51004

A parallelism \(\parallel\) on a projective space \(P\) is an equivalence relation on the set \(L\) of lines such that each point of \(P\) is incident with precisely one line from each equivalence class.
If a projective space \(P\) is endowed with two (not necessarily distinct) parallelisms, a left parallelism \(\parallel_l\) and a right parallelism \(\parallel_r\), then \((P, \parallel_l, \parallel_r)\) is said to be a projective double space (a concept introduced by H. Karzel et al. [J. Reine Angew. Math. 262–263, 153–157 (1973; Zbl 0265.50003)]) if it satisfies the following axiom:
(DS) For all triangles \(p_0p_1p_2\) in \(P\), there exists a common point of the line through \(p_2\) that is left parallel to the join of \(p_0\) and \(p_1\) and the line through \(p_1\) that is right parallel to the join of \(p_0\) and \(p_2\).
Given a projective double space \((P, \parallel_l, \parallel_r)\), each of \(\parallel_l\) and \(\parallel_r\) is referred to as a Clifford parallelism of \((P, \parallel_l, \parallel_r)\). A Clifford-like parallelism of \((P, \parallel_l, \parallel_r)\) is defined as a parallelism \(\parallel\) on \(P\) such that, for any two lines \(M\) and \(N\), \(M\parallel N\) implies \(M\parallel_l N\) or \(M\parallel_r N\).
All projective double spaces are isomorphic to certain algebraically defined projective spaces, the algebraic structures involved being built up from quaternion skew fields or from purely inseparable commutative field extensions of characteristic two. The paper under review deals with three-dimensional projective double spaces, as this is in the only dimension in which the phenomenon of proper Clifford parallelism exists (in all other dimensions, axiom (DS) implies \(\parallel_l=\parallel_r\)).
An algebraic description of three-dimensional projective double spaces, involves \(F\), a commutative field, \(H\), an \(F\)-algebra with unit \(1_H\), satisfying one of the following conditions:
(A)
\(H\) is a quaternion skew field with centre \(F1_H\).
(B)
\(H\) is a commutative field with degree \([H : F1_H] = 4\) and such that \(h^2 \in F1_H\) for all \(h \in H\).
The projective space \(P(H_F)\) on the vector space \(H_F\) is the set of all subspaces of \(H_F\) with incidence being symmetrised inclusion. Thus, points, lines, and planes of \(P(H_F)\) are the subspaces of \(H_F\) with vector dimension one, two, and three, respectively; left and right line parallelism is defined by \(M\parallel_l N\) if \(cM = N\) for some \(c \in H\setminus \{0\}\) and \(M\parallel_r N\) if \(Mc = N\) for some \(c \in H\setminus \{0\}\).
Here are the main results of this significant paper:
Let \((P(H_F), \parallel_l, \parallel_r)\) be a projective double space, where \(H\) is an \(F\)-algebra subject to (A) or (B). Let \(\parallel_l'\) and \(\parallel_r'\) be parallelisms such that \((P(H_F), \parallel_l', \parallel_r')\) is also a projective double space. Suppose that a parallelism \(\parallel\) of \((P(H_F)\) is Clifford-like with respect to both double space structures. Then, possibly up to a change of the attributes “left” and “right” in one of these double spaces, \(\parallel_l=\parallel_l'\) and \(\parallel_r=\parallel_r'\).
Let \(\parallel\) be a Clifford-like parallelism of \((P(H_F), \parallel_l, \parallel_r)\), where \(H\) is an \(F\)-algebra subject to (A) or (B). Then the following assertions are equivalent.
(a)
The parallelism \(\parallel\) is Clifford.
(b)
The parallelism \(\parallel\) admits a map \(\beta\) in the general semilinear group \(\Gamma L(H_F)\) that acts as a \(\parallel\)-preserving collineation on \((P(H_F)\), stabilises all its parallel classes, and acts as a non-identical collineation on the projective space \((P(H_F)\).

MSC:

51A15 Linear incidence geometric structures with parallelism
51J15 Kinematic spaces

Citations:

Zbl 0265.50003
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References:

[1] Bader, L.; Lunardon, G., Desarguesian spreads, Ric. Mat., 60, 1, 15-37 (2011) · Zbl 1223.51002
[2] Betten, A., Topalova, S., Zhelezova, S.: Parallelisms of \({\rm PG}(3,4)\) invariant under cyclic groups of order \(4\). In: M. Ćirić, M. Droste, J.-É. Pin (eds.) Algebraic informatics, Lecture Notes in Comput. Sci., vol. 11545, pp. 88-99. Springer, Cham. 8th International Conference, CAI 2019, Niš, Serbia (2019) · Zbl 1437.51002
[3] Betten, D.; Löwen, R., Compactness of the automorphism group of a topological parallelism on real projective \(3\)-space, Results Math., 72, 1-2, 1021-1030 (2017) · Zbl 1379.51003
[4] Betten, D.; Riesinger, R., Clifford parallelism: old and new definitions, and their use, J. Geom., 103, 1, 31-73 (2012) · Zbl 1258.51001
[5] Betten, D.; Riesinger, R., Collineation groups of topological parallelisms, Adv. Geom., 14, 1, 175-189 (2014) · Zbl 1291.51011
[6] Blunck, A.; Knarr, N.; Stroppel, B.; Stroppel, MJ, Clifford parallelisms defined by octonions, Monatsh. Math., 187, 3, 437-458 (2018) · Zbl 1400.51002
[7] Blunck, A.; Pasotti, S.; Pianta, S., Generalized Clifford parallelisms, Innov. Incidence Geom., 11, 197-212 (2010) · Zbl 1260.51001
[8] Cohn, PM, Basic Algebra (2003), London: Springer, London · Zbl 1003.00001
[9] Ellers, E.; Karzel, H., Kennzeichnung elliptischer Gruppenräume, Abh. Math. Sem. Univ. Hamburg, 26, 55-77 (1963) · Zbl 0109.39101
[10] Faith, CC, On conjugates in division rings, Canadian J. Math., 10, 374-380 (1958) · Zbl 0082.03203
[11] Giering, O., Vorlesungen über höhere Geometrie (1982), Braunschweig: Vieweg, Braunschweig · Zbl 0493.51001
[12] Havlicek, H., Durch Kollineationsgruppen bestimmte projektive Räume, Beiträge Algebra Geom., 27, 175-184 (1988) · Zbl 0677.51002
[13] Havlicek, H., On Plücker transformations of generalized elliptic spaces, Rend. Mat. Appl., 15, 1, 39-56 (1995) · Zbl 0828.51005
[14] Havlicek, H., A characteristic property of elliptic Plücker transformations, J. Geom., 58, 1-2, 106-116 (1997) · Zbl 0881.51019
[15] Havlicek, H., A note on Clifford parallelisms in characteristic two, Publ. Math. Debrecen, 86, 1-2, 119-134 (2015) · Zbl 1349.51002
[16] Havlicek, H., Clifford parallelisms and external planes to the Klein quadric, J. Geom., 107, 2, 287-303 (2016) · Zbl 1368.51001
[17] Havlicek, H., Pasotti, S., Pianta, S.: Clifford-like parallelisms. J. Geom. 110(1), 18 (2019a) · Zbl 1409.51002
[18] Havlicek, H., Pasotti, S., Pianta, S.: Automorphisms of a Clifford-like parallelism. Adv. Geom. (to appear). arXiv:1903.10331 (2019b) · Zbl 1461.51001
[19] Herzer, A.: Halbprojektive Translationsgeometrien. Mitt. Math. Sem. Giessen 127, i+136 pp. (1977) · Zbl 0363.50019
[20] Hirschfeld, JWP, Finite Projective Spaces of Three Dimensions (1985), Oxford: Oxford University Press, Oxford · Zbl 0574.51001
[21] Jacobson, N., Basic Algebra II (1989), New York: Freeman, New York · Zbl 0694.16001
[22] Johnson, NL, Parallelisms of projective spaces, J. Geom., 76, 1-2, 110-182 (2003) · Zbl 1036.51004
[23] Johnson, NL, Combinatorics of Spreads and Parallelisms, Pure and Applied Mathematics (Boca Raton) (2010), Boca Raton: CRC Press, Boca Raton · Zbl 1200.51001
[24] Karzel, H.; Kroll, H-J, Geschichte der Geometrie seit Hilbert (1988), Darmstadt: Wissenschaftliche Buchgesellschaft, Darmstadt · Zbl 0718.01003
[25] Karzel, H.; Maxson, CJ, Kinematic spaces with dilatations, J. Geom., 22, 2, 196-201 (1984) · Zbl 0537.51022
[26] Karzel, H.; Kroll, H-J; Sörensen, K., Invariante Gruppenpartitionen und Doppelräume, J. Reine Angew. Math., 262, 263, 153-157 (1973) · Zbl 0265.50003
[27] Karzel, H.; Kroll, H-J; Sörensen, K., Projektive Doppelräume, Arch. Math. Basel, 25, 206-209 (1974) · Zbl 0296.50003
[28] Kroll, H-J, Bestimmung aller projektiven Doppelräume, Abh. Math. Sem. Univ. Hamburg, 44, 139-142 (1975) · Zbl 0317.50005
[29] Lam, TY, A First Course in Noncommutative Rings. Graduate Texts in Mathematics (2001), New York: Springer, New York · Zbl 0980.16001
[30] Lam, TY, Introduction to Quadratic Forms Over Fields, Graduate Studies in Mathematics (2005), Providence: American Mathematical Society, Providence · Zbl 1068.11023
[31] Löwen, R., Compactness of the automorphism group of a topological parallelism on real projective \(3\)-space: the disconnected case, Bull. Belg. Math. Soc. Simon Stevin, 25, 4, 629-640 (2018) · Zbl 1436.51011
[32] Löwen, R., A characterization of Clifford parallelism by automorphisms, Innov. Incidence Geom., 17, 1, 43-46 (2019) · Zbl 1406.51010
[33] Löwen, R., Parallelisms of PG \((3,{\mathbb{R}})\) admitting a \(3\)-dimensional group, Beiträge Algebra Geom., 60, 2, 333-337 (2019) · Zbl 1417.51013
[34] Lüneburg, H., Translation Planes (1980), Berlin: Springer, Berlin · Zbl 0446.51003
[35] Marchi, M.; Perelli Cippo, C., Su una particolare classe di \(S\)-spazi, Rend. Semin. Mat. Brescia, 4, 3-42 (1980) · Zbl 0458.51001
[36] Pianta, S., On automorphisms for some fibered incidence groups, J. Geom., 30, 2, 164-171 (1987) · Zbl 0631.51009
[37] Pianta, S.; Zizioli, E., Collineations of geometric structures derived from quaternion algebras, J. Geom., 37, 1-2, 142-152 (1990) · Zbl 0709.51019
[38] Seier, W., Kollineationen von Translationsstrukturen, J. Geom., 1, 2, 183-195 (1971) · Zbl 0227.50006
[39] Seier, W., Isomorphismen verallgemeinerter Parallelstrukturen, J. Geom., 3, 2, 165-178 (1973) · Zbl 0265.50014
[40] Tyrrell, J.A., Semple, J.G.: Generalized Clifford Parallelism. Cambridge Tracts in Mathematics and Mathematical Physics, No. 61. Cambridge University Press, London, New York (1971) · Zbl 0213.22102
[41] van Buggenhaut, J., Principe de trialité et parallélisme dans l’espace elliptique à \(7\) dimensions, Acad. R. Belg. Bull. Cl. Sci., 5, 54, 577-584 (1968) · Zbl 0176.17902
[42] van Buggenhaut, J., Algèbres d’octaves et parallélisme dans l’espace elliptique à \(7\) dimensions, Acad. R. Belg. Bull. Cl. Sci., 5, 54, 662-670 (1968) · Zbl 0167.18901
[43] van Buggenhaut, J., Deux généralisations du parallélisme de Clifford, Bull. Soc. Math. Belg., 20, 406-412 (1968) · Zbl 0176.17903
[44] Vaney, F., Le parallélisme absolu dans les espaces elliptiques réels à \(3\) et \(7\) dimensions et le principe de trialité dans l’espace elliptique à \(7\) dimensions (1929), Thèse: Université de Paris, Thèse
[45] Wähling, H., Darstellung zweiseitiger Inzidenzgruppen durch Divisionsalgebren, Abh. Math. Sem. Univ. Hamburg, 30, 220-240 (1967) · Zbl 0195.22201
[46] Wähling, H., Konjugierte Teilkörper eines Körpers, Arch. Math. (Basel), 37, 1, 52-58 (1981) · Zbl 0493.16014
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