Translation plane

Template:Short description 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 Template:Math represent a point, and Template:Math represent a line. A central collineation with center Template:Math and axis Template:Math is a collineation fixing every point on Template:Math and every line through Template:Math. It is called an elation if Template:Math is on Template:Math, otherwise it is called a homology. The central collineations with center Template:Math and axis Template:Math form a group. A line Template:Math in a projective plane Template:Math is a translation line if the group of all elations with axis Template:Math acts transitively on the points of the affine plane obtained by removing Template:Math from the plane Template:Math, Template:Math (the affine derivative of Template:Math). 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 $\cdot$ (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 $(a,b)$ where $a$ and $b$ are elements of the quasifield, and the lines are the sets of points $(x,y)$ satisfying an equation of the form $y=m\cdot x+b$ , as $m$ and $b$ vary over the elements of the quasifield, together with the sets of points $(x,y)$ satisfying an equation of the form $x=a$ , as $a$ 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 Template:Math, where $n\ge 1$ is an integer and Template:Math a division ring, is a partition of the space into pairwise disjoint Template:Math-dimensional subspaces. In the finite case, a spread of Template:Math is a set of Template:Math Template:Math-dimensional subspaces, with no two intersecting.

Given a spread Template:Math of Template:Math, the Bruck-Bose construction produces a translation plane as follows: Embed Template:Math as a hyperplane $\mathrm{\Sigma }$ of Template:Math. Define an incidence structure Template:Math with "points," the points of Template:Math not on $\mathrm{\Sigma }$ and "lines" the Template:Math-dimensional subspaces of Template:Math meeting $\mathrm{\Sigma }$ in an element of Template:Math. Then Template:Math is an affine translation plane. In the finite case, this procedure produces a translation plane of order Template:Math.

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 Template:Math (Template:Math is necessarily a division ring) can be generated from a spread of Template:Math using the Bruck-Bose construction, where Template:Math 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 Template:Math be a Template:Math-dimensional vector space over a field Template:Math. A spread of Template:Math is a set Template:Math of Template:Math-dimensional subspaces of Template:Math that partition the non-zero vectors of Template:Math. The members of Template:Math are called the components of the spread and if Template:Math and Template:Math are distinct components then Template:Math. Let Template:Math be the incidence structure whose points are the vectors of Template:Math and whose lines are the cosets of components, that is, sets of the form Template:Math where Template:Math is a vector of Template:Math and Template:Math is a component of the spread Template:Math. Then:

Template:Math is an affine plane and the group of translations Template:Math for Template:Math in Template:Math is an automorphism group acting regularly on the points of this plane.

The finite case

Let Template:Math, the finite field of order Template:Math and Template:Math the Template:Math-dimensional vector space over Template:Math represented as:

$V=\{(x,y):x,y\in F^n\}.$

Let Template:Math be Template:Math matrices over Template:Math with the property that Template:Math is nonsingular whenever Template:Math. For Template:Math define,

$V_i=\{(x,x{M}_i):x\in F^n\},$

usually referred to as the subspaces "Template:Math". Also define:

$V_{q^n}=\{(0,y):y\in F^n\},$

the subspace "Template:Math".

The set Template:Math} is a spread of Template:Math.

The set of matrices Template:Math used in this construction is called a spread set, and this set of matrices can be used directly in the projective space $PG(2n-1,q)$ to create a spread in the geometric sense.

Reguli and regular spreads

Let $\mathrm{\Sigma }$ be the projective space Template:Math for $n\ge 1$ an integer, and Template:Math a division ring. A regulus Template:Math in $\mathrm{\Sigma }$ is a collection of pairwise disjoint Template:Math-dimensional subspaces with the following properties:

1. Template:Math contains at least 3 elements
2. Every line meeting three elements of Template:Math, called a transversal, meets every element of Template:Math
3. Every point of a transversal to Template:Math lies on some element of Template:Math

Any three pairwise disjoint Template:Math-dimensional subspaces in $\mathrm{\Sigma }$ lie in a unique regulus. A spread Template:Math of $\mathrm{\Sigma }$ is regular if for any three distinct Template:Math-dimensional subspaces of Template:Math, all the members of the unique regulus determined by them are contained in Template:Math. For any division ring Template:Math with more than 2 elements, if a spread Template:Math of Template:Math is regular, then the translation plane created by that spread via the André/Bruck-Bose construction is a Moufang plane. A slightly weaker converse holds: if a translation plane is Pappian, then it can be generated via the André/Bruck-Bose construction from a regular spread.

In the finite case, Template:Math must be a field of order $q>2$, and the classes of Moufang, Desarguesian and Pappian planes are all identical, so this theorem can be refined to state that a spread Template:Math of Template:Math is regular if and only if the translation plane created by that spread via the André/Bruck-Bose construction is Desarguesian.

In the case where Template:Math is the field $GF(2)$, all spreads of Template:Math are trivially regular, since a regulus only contains three elements. While the only translation plane of order 8 is Desarguesian, there are known to be non-Desarguesian translation planes of order Template:Math for every integer $e\ge 4$.

Families of non-Desarguesian translation planes

• Hall planes - constructed via Bruck/Bose from a regular spread of $PG(3,q)$ where one regulus has been replaced by the set of transversal lines to that regulus (called the opposite regulus).
• Subregular planes - constructed via Bruck/Bose from a regular spread of $PG(3,q)$ where a set of pairwise disjoint reguli have been replaced by their opposite reguli.
• André planes
• Nearfield planes
• Semifield planes

Finite translation planes of small order

It is well known that the only projective planes of order 8 or less are Desarguesian, and there are no known non-Desarguesian planes of prime order. Finite translation planes must have prime power order. There are four projective planes of order 9, of which two are translation planes: the Desarguesian plane, and the Hall plane. The following table details the current state of knowledge:

Order	Number of Non-Desarguesian Translation Planes
9	1
16	7
25	20
27	6
32	≥8
49	1346
64	≥2833

References

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Further reading

• Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, Marcel Dekker Template:Isbn .

External links

• Lecture Notes on Projective Geometry
• Publications of Keith Mellinger