Translation planes. Foundations and construction principles
By definition a translation plane is an affine plane where the group of translations acts transitively on the set of points, and these planes are exactly the coordinate planes over quasifields. The interest in translation planes increased very much with J. André's representation theorem of 1954: There is a one to one correspondence between translation planes and spreads \((V, {\mathfrak B})\) (or congruences as André called them), that is a vector space \((V, K)\) together with a subset \(\mathfrak B\) of vector subspaces such that \(\bigcup {\mathfrak B} = V\) and \(\forall A, B \in {\mathfrak B}\), \(A \neq B : V = A \oplus B\). In the case \(\dim (V,K) = 4\) one has the following projective translation: A spread of a 3-dimensional projective space \((P, {\mathfrak L})\) is a subset \(\mathfrak B\) of the line set \(\mathfrak L\) such that \(\forall x \in P\), \(\exists_1 B \in {\mathfrak B} : x \in B\). In this lecture note the author gives an excellent survey on this fascinating area, which can serve as a text for a lecture, a seminar, for privat study but also has a basis for further research. The content is described by the titles: I. Foundations; II. Spreads of 3-dimensional projective spaces; III. Kinematic spaces; IV. Examples and supplements; V. Locally compact 4-dimensional translation planes; VI. Planes of Lenz type V with complex kernel; VII. Locally compact translation planes of higher dimension.
scientific article; zbMATH DE number 803276