In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation.
In a projective plane, let represent a point, and represent a line. A central collineation with center and axis is a collineation fixing every point on and every line through . It is called an elation if is on , otherwise it is called a homology. The central collineations with center and axis form a group. A line in a projective plane is a translation line if the group of all elations with axis acts transitively on the points of the affine plane obtained by removing from the plane , (the affine derivative of ). A projective plane with a translation line is called a translation plane.
The affine plane obtained by removing the translation line is called an affine translation plane. While it is often easier to work with projective planes, in this context several authors use the term translation plane to mean affine translation plane.
Algebraic construction with coordinates
Every projective plane can be coordinatized by at least one planar ternary ring. For translation planes, it is always possible to coordinatize with a quasifield. However, some quasifields satisfy additional algebraic properties, and the corresponding planar ternary rings coordinatize translation planes which admit additional symmetries. Some of these special classes are:
Nearfield planes - coordinatized by nearfields. Semifield planes - coordinatized by semifields, semifield planes have the property that their dual is also a translation plane. Moufang planes - coordinatized by alternative division rings, Moufang planes are exactly those translation planes that have at least two translation lines. Every finite Moufang plane is Desarguesian and every Desarguesian plane is a Moufang plane, but there are infinite Moufang planes that are not Desarguesian (such as the Cayley plane).
Given a quasifield with operations + (addition) and (multiplication), one can define a planar ternary ring to create coordinates for a translation plane. However, it is more typical to create an affine plane directly from the quasifield by defining the points as pairs where and are elements of the quasifield, and the lines are the sets of points satisfying an equation of the form , as and vary over the elements of the quasifield, together with the sets of points satisfying an equation of the form , as varies over the elements of the quasifield.
Geometric construction with spreads (Bruck/Bose)
Translation planes are related to spreads of odd-dimensional projective spaces by the Bruck-Bose construction. A spread of , where is an integer and a division ring, is a partition of the space into pairwise disjoint -dimensional subspaces. In the finite case, a spread of is a set of -dimensional subspaces, with no two intersecting.
Given a spread of , the Bruck-Bose construction produces a translation plane as follows: Embed as a hyperplane of . Define an incidence structure with "points," the points of not on and "lines" the -dimensional subspaces of meeting in an element of . Then is an affine translation plane. In the finite case, this procedure produces a translation plane of order .
The converse of this statement is almost always true. Any translation plane which is coordinatized by a quasifield that is finite-dimensional over its kernel ( is necessarily a division ring) can be generated from a spread of using the Bruck-Bose construction, where is the dimension of the quasifield, considered as a module over its kernel. An instant corollary of this result is that every finite translation plane can be obtained from this construction.
Algebraic construction with spreads (André) André gave an earlier algebraic representation of (affine) translation planes that is fundamentally the same as Bruck/Bose. Let be a -dimensional vector space over a field . A spread of is a set of -dimensional subspaces of that partition the non-zero vectors of . The members of are called the components of the spread and if and are distinct components then . Let be the incidence structure whose points are the vectors of and whose lines are the cosets of components, that is, sets of the form where is a vector of and is a component of the spread . Then: is an affine plane and the group of translations for in is an automorphism group acting regularly on the points of this plane.