Automorphisms of a Clifford-like parallelism

Abstract

In this paper we focus on the description of the automorphism group ${\Gamma }_{\parallel }$ of a Clifford-like parallelism $\parallel$ on a $3$-dimensional projective double space $(\mathbb{P}(H_F),{\parallel }_{\ell },{\parallel }_r)$ over a quaternion skew field $H$ (with centre a field $F$ of any characteristic). We compare ${\Gamma }_{\parallel }$ with the automorphism group ${\Gamma }_{\ell }$ of the left parallelism ${\parallel }_{\ell }$, which is strictly related to $\mathrm{Aut}(H)$. We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose $\parallel$ in such a way that ${\Gamma }_{\parallel }$ is either properly contained in ${\Gamma }_{\ell }$ or coincides with ${\Gamma }_{\ell }$ even though $\parallel \ne {\parallel }_{\ell }$.

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