Clifford-like parallelisms

Abstract

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

Characterising Clifford parallelisms among Clifford-like parallelisms

“…both double space structures then the given double spaces coincide up to a change of the attributes "left" and "right" in one of them. This finding improves [17,Thm. 4.15] (see Corollary 4.2) and it simplifies matters considerably.…”

“…It is a widely used strategy in mathematics to define a new structure by modifying a given one. The definition of a Clifford-like parallelism from [7] and [17], which is recalled in Section 2, follows these lines. The starting point is a projective double space (P, , r ), that is, a projective space P together with a left parallelism and a right parallelism r on its line set such that the so-called double space axiom (DS) is satisfied.…”

Characterising Clifford parallelisms among Clifford-like parallelisms

“…The symbol denotes the automorphism group of . This terminology is in accordance with the one in Havlicek (2019b).…”

“…4.15). Therefore, the rather technical proof in Havlicek et al (2019a), which relies on H being a quaternion skew field, can now be avoided.…”

“…It is a widely used strategy in mathematics to define a new structure by modifying a given one. The definition of a Clifford-like parallelism from Blunck et al (2010) and Havlicek et al (2019a), which is recalled in Sect. 2, follows these lines.…”