Die affinen Projektivitätengruppen der lokalkompakten zusammenhängenden Translationsebenen. (The affine groups of projectivities of locally compact connected translation planes)
Let \({\mathcal U}\) be a locally compact connected translation plane. If \({\mathcal U}\) is a Moufang plane (in which case it is coordinatized by the reals, the complexes, the quaternions, or the Cayley numbers), then the group \(\Pi^{aff}({\mathcal U})\) of affine projectivities has been determined. [See, for example, the first author, Geom. Dedicata 13, 435- 438 (1983; Zbl 0523.51006), or the second author, Math. Z. 155, 11-21 (1977; Zbl 0354.50004).] In this article the authors consider the case where \({\mathcal U}\) is not a Moufang plane and prove that \(\Pi^{aff}({\mathcal U})\), as a topological transformation group, is isomorphic to the connected component \(AGL(n,K)^ 1\) of the affine group of \(K^ n\), where K is the kernel of \({\mathcal U}\) and 2n is the dimension of \({\mathcal U}\) over K. In the process of proving the result given above, the authors characterize the quaternions among the four dimensional locally compact quasifields containing \({\mathbb{C}}\) in their kernel. The authors also give a necessary and sufficient condition involving \(\Pi^{aff}({\mathcal U})\) for a locally compact connected affine plane to be a translation plane. This sharpens a result of the second author.
locally compact connected translation plane
scientific article; zbMATH DE number 3990224