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Karzel, Helmut; Kroll, Hans-Joachim; S�rensen, Kay

doi:10.1007/bf01238665

Cited by 10 publications

(6 citation statements)

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“…The projective double spaces (P, ℓ , r ) with ℓ r are precisely the ones that can be obtained from quaternion skew fields (see [17], [18], [15], [22]). A detailed account is the topic of the next section.…”

Section: Blends Of Parallelismsmentioning

confidence: 99%

“…Generalising this situation, H. Karzel, H.-J. Kroll and K. Sörensen in 1973 introduced the notion of a projective double space (P, ℓ , r ), that is a projective space (of unspecified dimension, over an unspecified field) equipped with two parallelism relations fulfilling a configurational property which can be expressed by the axiom (DS) of Section 3 (see [17], [18]). The real projective 3-space with left and right Clifford parallelisms is an example and it turns out that the projective double spaces (P, ℓ , r ) with ℓ r are necessarily of dimension 3 and precisely the ones that can be obtained from a quaternion skew field H over a field F as in Section 4 (see [17], [18], [15], [22]).…”

Section: Introductionmentioning

confidence: 99%

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Clifford-like parallelisms

2018

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In this paper we focus on the description of the automorphism group Γ of a Clifford-like parallelism on a 3-dimensional projective double space P(H F ), ℓ , r over a quaternion skew field H (with centre a field F of any characteristic). We compare Γ with the automorphism group Γ ℓ of the left parallelism ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose in such a way that Γ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ℓ .Mathematics Subject Classification (2010): 51A15, 51J15

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Section: Blends Of Parallelismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Clifford-like parallelisms

2018

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“…The point Fe and the parallelisms and r can be used to make the point set P(H F ) into a two-sided incidence group with unit element Fe (Karzel et al 1973, §3). (The prism axiom appearing in Karzel et al (1973) can be avoided (Karzel et al 1974, Satz 1), (Kroll 1975, Satz 2).) Then, using the group structure on P(H F ), the F-vector space H can be endowed with a multiplication making it into a field with unit element e (see Ellers andKarzel 1963, Satz 1 andWähling 1967, Hauptsatz).…”

Section: Remark 21mentioning

confidence: 97%

Characterising Clifford parallelisms among Clifford-like parallelisms

2020

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We recall the notions of Clifford and Clifford-like parallelisms in a 3-dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.

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“…[18, p. 75-78], [19]) which is also a prism space (cf. [19,20]) as well as a kinematic space which is defined as follows.…”

Section: Proof Putmentioning

confidence: 99%

“…By [19], (1) kinematic spaces and prism spaces coincide and (2) every prism space with = r is a kinematic derivation of a (general) quaternion skew field. By [20], (1) any two different left parallel lines of a projective double space span a 3-dimensional projective double space, (2) any 3-dimensional projective double space is a prism space, and (3) there exist no finite projective double spaces. By [17], every projective double space with distinct left and right parallelisms has dimension 3.…”

Section: Proof Putmentioning

confidence: 99%

Clifford parallelism: old and new definitions, and their use

Betten

Riesinger²

2012

J. Geom.

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Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space PG(3, R) whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein's definition). Our new access to the topic 'Clifford parallelism' is free of complexification and involves Klein's correspondence λ of line geometry together with a bijective map γ from all regular spreads of PG(3, R) onto those lines of PG(5, R) having no common point with the Klein quadric; a regular parallelism P of PG(3, R) is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of PG(3, R) are Clifford (D3 = dimensionality definition). Submission of (D2) to λ −1 yields a complexification free definition of a Clifford parallelism which uses only elements of PG(3, R): A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group Aute(P C ) of all automorphic collineations and dualities of the Clifford parallelism P C and show Aute(P C ) ∼ = (SO3R × SO3R) Z2.Mathematics Subject Classification (2010). 51A15, 51M30, 51H10.Keywords. Topological parallelism, complexification, focal line of a regular spread, imaginary quadric, Klein's correspondence of line geometry, elliptic displacement.

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Projektive Doppelr�ume

Cited by 10 publications

References 0 publications

Clifford-like parallelisms

Clifford-like parallelisms

Characterising Clifford parallelisms among Clifford-like parallelisms

Clifford parallelism: old and new definitions, and their use

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