Clifford parallelisms and external planes to the Klein quadric

Abstract

For any three-dimensional projective space P(V), where V is a vector space over a field F of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of P(V) and those planes of P(V ∧ V) that are external to the Klein quadric representing the lines of P(V). We also give two characterisations of a Clifford parallelism of P(V), both of which avoid the ambient space of the Klein quadric.Mathematics Subject Classification (2010): 51A15, 51J15

Pencilled regular parallelisms

“…Furthermore, by the action of π 5 on the lattice of subspaces of PG(5, K), we obtain span λ(C) | C ∈ P = S ⊂ P 5 | S is a solid and π 5 (κ 1 ) ⊂ S . (14) This description of P in terms of the Klein correspondence coincides with the definition of a parallelism in [16,Def. 4.2], which relies on the choice of an external plane to H 5 ; in our context this distinguished external plane is π 5 (κ 1 ).…”

“…We add in passing that our proof of the proposition above uses [16,Thm. 4.8], which in turn is based upon a series of other results about Clifford parallelism.…”

“…These two parallelisms turn PG(3, K) into a projective double space; they coincide precisely when (B) applies. Note also that (B) implies that the characteristic of K is two and that H is a purely inseparable extension of K. More generally, a parallelism P of an arbitrary projective space PG(3, K) is said to be Clifford if the underlying vector space of PG(3, K) can be made into a K-algebra H, subject to (A) or (B), in such a way that P coincides with the left or right parallelism arising from H [16,Def. 3.4].…”

“…3.4]. We refer to [7], [10], [11], [13], [15], [16], [19, pp. 112-115], [21, § 14] and [24] for surveys, recent results, and a wealth of references on Clifford parallelism.…”

Characterising Clifford parallelisms among Clifford-like parallelisms

“…There are various other ways to define a Clifford parallelism on a threedimensional (necessarily pappian) projective space. We refer to (Betten and Riesinger 2012;Blunck et al 2010), (Havlicek 1995, p. 46), (Havlicek 1997, Sect. 2), Havlicek (2015Havlicek ( , 2016 and the references given there. On that account, it is our aim to make use only of the above algebraic approach.…”