Characterising Clifford parallelisms among Clifford-like parallelisms (Q2022352)

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 \textit{projective double space} (a concept introduced by \textit{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 \textit{Clifford parallelism} of \((P, \parallel_l, \parallel_r)\). A \textit{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: \begin{itemize} \item[(A)] \(H\) is a quaternion skew field with centre \(F1_H\). \item[(B)] \(H\) is a commutative field with degree \([H : F1_H] = 4\) and such that \(h^2 \in F1_H\) for all \(h \in H\). \end{itemize} 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. \begin{itemize} \item[(a)] The parallelism \(\parallel\) is Clifford. \item[(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 \((P(H_F)\). \end{itemize}