CROSS-RATIOS IN MOUFANG-KLINGENBERG PLANES
ABSVRACT. Certain geometric groups operating on a line g in a Moufang-Klingenberg plane J/ are described algebraically in terms of the underlying alternative ring R. For the case of the dual numbers R = A + Ae (A alternative field, e2= 0) a notion of cross-ratio is introduced on the line. We establish some connections between the geometric groups and the cross-ratio which are well known from classical projective planes.
In this note, we want to investigate geometrically defined permutation groups of a line g in a so-called cohesive Moufang-Klingenberg plane JΒ’. This plane can be eoordinatized by an alternative chain ring R. In Section 1 we describe the geometric permutations algebraically with the help of certain Jordan automorphisms of R.
If in R the conjugacy relation is transitive, we may introduce a notion of cross-ratio on the line generalizing the usual one (cf. [5], [15]). In Section 2 the interrelations between our groups on g and the cross-ratio are analysed. Unfortunately, only in the case of the dual numbers A(e) = A + As (5 2 = 0) over an alternative field A we can prove that the projectivities of g preserve crossratios. So in Section 2 we restrict ourselves to the MK-plane over A(e). We show certain transitivity theorems for the geometric groups, and that -in the case that A is not associative -every permutation preserving cross-ratios is geometrically induced. By this, we generalize A. Schleiermacher's results on cross-ratios in proper Moufang planes of char Β’ 2 ([15], also cf. [6], [9]).
1. GEOMETRIC PERMUTATIONS
In this section we consider arbitrary cohesive Moufang-Klingenberg planes: DEFINITION. Let J{ = (~, ~, E, czzz) consist of an incidence structure (~, 5Β’, ~) (points, lines, incidence) and an equivalence relation 'czz' (neighbouring) on ~ and on ~v. Then J{ is called a projective Klingenberg plane (PK-plane), if it satisfies the following axioms: (PK1) Any two non-neighbouring points P, Q ~ ~ have exactly one joining line ge~ (i.e. Peg and Q~g), denoted by g = PQ. (PK2) Any two non-neighbouring lines g, h ~ o~ have exactly one intersection point P ~ ~ (i.e. P ~ g and P ~ h), denoted by P = g c~ h.