In mathematics, a translation plane is a particular kind of projective plane. Almost all non-Desarguesian planes are either translation planes or are related to this type of incidence structure. 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 centre and axis form a group. A line in a projective plane is a translation line if the group of elations with axis acts transitively on the points of the affine plane obtained by removing from the plane, . A projective plane with a translation line is called a translation plane and the affine plane obtained by removing the translation line is called an affine translation plane. While in general it is often easier to work with projective planes, in this context the affine planes are preferred and several authors simply use the term translation plane to mean affine translation plane.
Moufang planes
A translation plane having at least two translation lines is a Moufang plane. All the lines of a Moufang plane are 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. Moufang planes are coordinatized by alternative division rings.
Relationship to (geometric) spreads
Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction. A spread of is a set of lines, with no two intersecting. Equivalently, it is a partition of the points of into lines. Given a spread of, the André/Bruck-Bose construction produces a translation plane of order as follows: Embed as a hyperplane of. Define an incidence structure with "points," the points of not on and "lines" the planes of meeting in a line of. Then is a translation affine plane of order. Let be the projective completion of.
Reguli and regular spreads
In a set of mutually skew lines with the property that any line intersecting three lines of must intersect all the lines of is called a regulus. The lines intersecting all the lines of are called transversals of. Any three mutually skew lines of lie in precisely one regulus. A spread of is regular if for any three distinct lines of all the lines of the unique regulus determined by them are contained in.
For, a spread of is regular if and only if the translation plane defined by that spread is desarguesian.
All spreads of are regular. If the geometry is defined over an infinite field, the same result holds but "desarguesian" must be replaced by "pappian".
An algebraic representation of translation planes can be obtained as follows: 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:
Finite construction
Let, the finite field of order and the -dimensional vector space over represented as: Let be matrices over with the property that is nonsingular whenever. For define, usually referred to as the subspaces "". Also define: the subspace "". The matrices used in this construction are called spread matrices or slope matrices.
Examples of regular spreads
A regular spread may be constructed in the following way. Let be a field and an -dimensional extension field of. Let considered as a -dimensional vector space over. The set of all 1-dimensional subspaces of over is a regular spread of. In the finite case, the field can be represented as a subring of the matrices over. With respect to a fixed basis of over, the multiplication maps, for in, are -linear transformations and can be represented by matrices over. These matrices are the spread matrices of a regular spread. As a specific example, the following nine matrices represent as 2 × 2 matrices over and so provide a spread set of.
Modifying spread sets
The set of transversals of a regulus also form a regulus, called the opposite regulus of. If a spread of contains a regulus, the removal of and replacing it by its opposite regulus produces a new spread. This process is a special case of a more general process called derivation or net replacement. Starting with a regular spread of and deriving with respect to any regulus produces a Hall plane. In more generality, the process can be applied independently to any collection of reguli in a regular spread, resulting in André planes.
