Translation plane

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.^[1]

In a projective plane, let $P$ represent a point, and $l$ represent a line. A central collineation with center $P$ and axis $l$ is a collineation fixing every point on $l$ and every line through $P$ . It is called an elation if $P$ is on $l$ , otherwise it is called a homology. The central collineations with centre $P$ and axis $l$ form a group.^[2]

A line $l$ in a projective plane $Π$ is a translation line if the group of elations with axis $l$ acts transitively on the points of the affine plane obtained by removing $l$ from the plane $Π$ , $Π l$ (the affine derivative of $Π$ ). 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.^[3]^[4]

Moufang planes

Main article: Moufang plane

A (projective) translation plane having at least three nonconcurrent translation lines is a Moufang plane.^[5] 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 (such as the Cayley plane). 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.^[6] A spread of $PG(3, q)$ is a set of $q 2 + 1$ lines, with no two intersecting. Equivalently, it is a partition of the points of $PG(3, q)$ into lines.

Given a spread $S$ of $PG(3, q)$ , the André/Bruck-Bose construction produces a translation plane $π(S)$ of order $q 2$ as follows: Embed $PG(3, q)$ as a hyperplane of $PG(4, q)$ . Define an incidence structure $A (S)$ with "points," the points of $PG(4, q)$ not on $PG(3, q)$ and "lines" the planes of $PG(4, q)$ meeting $PG(3, q)$ in a line of $S$ . Then $A (S)$ is a translation affine plane of order $q 2$ . Let $π(S)$ be the projective completion of $A (S)$ .^[7] ^[8]

Reguli and regular spreads

In $PG(3, q)$ a set $R$ of $q + 1$ mutually skew lines with the property that any line intersecting three lines of $R$ must intersect all the lines of $R$ is called a regulus. The lines intersecting all the lines of $R$ are called transversals of $R$ . Any three mutually skew lines of $PG(3, q)$ lie in precisely one regulus.

A spread $S$ of $PG(3, q)$ is regular if for any three distinct lines of $S$ all the lines of the unique regulus determined by them are contained in $S$ .^[9]

For $q > 2$ , a spread $S$ of $PG(3, q)$ is regular if and only if the translation plane defined by that spread is desarguesian.

All spreads of $PG(3, 2)$ are regular. If the geometry is defined over an infinite field, the same result holds but "desarguesian" must be replaced by "pappian".^[10]

Algebraic representation

An algebraic representation of (affine) translation planes can be obtained as follows: Let $V$ be a $2 n$ -dimensional vector space over a field $F$ . A spread of $V$ is a set $S$ of $n$ -dimensional subspaces of $V$ that partition the non-zero vectors of $V$ . The members of $S$ are called the components of the spread and if $V i$ and $V j$ are distinct components then $V i \oplus V j = V$ . Let $A$ be the incidence structure whose points are the vectors of $V$ and whose lines are the cosets of components, that is, sets of the form $v + U$ where $v$ is a vector of $V$ and $U$ is a component of the spread $S$ . Then:^[11]

A

is an affine plane and the group of translations

x \to x + w

for

w

V

is an automorphism group acting regularly on the points of this plane.

Finite construction

Let $F = GF(q) = F q$ , the finite field of order $q$ and $V$ the $2 n$ -dimensional vector space over $F$ represented as:

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

Let $M 0, M 1, ..., M q n - 1$ be $n \times n$ matrices over $F$ with the property that $M i - M j$ is nonsingular whenever $i \neq j$ . For $i = 0, 1, ..., q n - 1$ define,

V_{i}=\{(x,xM_{i})\colon x\in F^{n}\},

usually referred to as the subspaces " $y = xM i$ ". Also define:

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

the subspace " $x = 0$ ".

The set

{V 0, V 1, ..., V q n

} is a spread of

V

The matrices $M i$ 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 $F$ be a field and $E$ an $n$ -dimensional extension field of $F$ . Let $V = E 2$ considered as a $2 n$ -dimensional vector space over $F$ . The set of all 1-dimensional subspaces of $V$ over $E$ (and hence, $n$ -dimensional over $F$ ) is a regular spread of $V$ .

In the finite case, the field $E = GF(q n)$ can be represented as a subring of the $n \times n$ matrices over $F = GF(q)$ . With respect to a fixed basis of $E$ over $F$ , the multiplication maps, $x \to αx$ for $α$ in $E$ , are $F$ -linear transformations and can be represented by $n \times n$ matrices over $F$ . These matrices are the spread matrices of a regular spread.^[12]

As a specific example, the following nine matrices represent $GF(9)$ as 2 × 2 matrices over $GF(3)$ and so provide a spread set of $AG(2, 9)$ .

\left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}1&1\\2&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\1&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right].

Modifying spread sets

The set of transversals of a regulus $R$ also form a regulus, called the opposite regulus of $R$ . If a spread $S$ of $PG(3, q)$ contains a regulus $R$ , the removal of $R$ and replacing it by its opposite regulus produces a new spread $S *$ . This process is a special case of a more general process called derivation or net replacement.^[13]

Starting with a regular spread of $PG(3, q)$ 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.

Notes

↑ Projective Planes On projective planes
↑ Geometry Translation Plane Retrieved on June 13, 2007
↑ Hughes & Piper 1973, p. 100
↑ Johnson, Jha & Biliotti 2007, p. 5
↑ Hughes & Piper 1973, p. 101
↑ Ball, Simeon; John Bamberg; Michel Lavrauw; Tim Penttila (2003-09-15). "Symplectice Spreads" (PDF). Polytechnic University of Catalonia. Retrieved 2008-10-08.
↑ André, Johannes (1954), "Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe", Math. Zeitschr., 60: 156–186
↑ Bruck, R. H.; Bose, R. C. (1964), "The Construction of Translation Planes from Projective Spaces", Journal of Algebra, 1: 85–102, doi:10.1016/0021-8693(64)90010-9
↑ F. A. Scherk & Günther Pabst (1977) "Indicator sets, reguli, and a new class of spreads", page 144, Canadian Journal of Mathematics 29(1): 132–54
↑ Dembowski 1968, p. 221, see especially footnote 1).
↑ Moorhouse 2007, p. 13
↑ Moorhouse 2007, p. 15
↑ Johnson, Jha & Biliotti 2007, p. 49

References

Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275
Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, Springer-Verlag, ISBN 0-387-90044-6
Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of Finite Translation Planes, Chapman&Hall/CRC, ISBN 978-1-58488-605-1
Lüneburg, Heinz (1980), Translation Planes, Berlin: Springer Verlag, ISBN 0-387-09614-0
Moorhouse, Eric (2007), Incidence Geometry (PDF)

External links

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