translation plane

Let $\pi$ be a projective plane. Recall that a central collineation on $\pi$ is a collineation $\rho$ with a center $C$ and an axis $\mathrm{\ell }$. It is well-known that $C$ and $\mathrm{\ell }$ are uniquely determined. We also call $\rho$ a $(C,\mathrm{\ell })$-collineation.

Definition. Let $\pi$ be a projective plane. We say that $\pi$ is $\mathrm{(}C\mathrm{,}\mathrm{\ell }\mathrm{)}$-transitive if there is a point $C$ and a line $\mathrm{\ell }$, such that for any points $P,Q$ where

• $P,Q$ and $C$ are collinear and pairwise distinct,

• $P,Q\notin \mathrm{\ell }$,

there is a $(C,\mathrm{\ell })$-collineation $\rho$ such that $\rho (P)=Q$.

It can be shown that $\pi$ if $(C,\mathrm{\ell })$-transitive iff it is $(C,\mathrm{\ell })$-Desarguesian; that is, if two triangles are perspective from point $C$, then they are perspective from line $\mathrm{\ell }$. From this, it is easy to see that $\pi$ is a Desarguesian plane iff it is $(C,\mathrm{\ell })$-transitive for any point $C$ and any line $\mathrm{\ell }$, of $\pi$.

Now, suppose that $C$ lies on $\mathrm{\ell }$. Then one can show that $\pi$ is $(C,\mathrm{\ell })$-transitive iff it can be coordinatized by a linear ternary ring $R$ such that $R$ is a group with respect to the derived operation $+$ (addition). When $\pi$ is so coordinatized, $\mathrm{\ell }$ is the line at infinity, and $C$ is the point whose coordinate is $(\mathrm{\infty })$.

This group is not necessarily abelian. So what condition(s) must be imposed on $\pi$ so that $(R,+)$ is an abelian group? The answer lies in the next definition:

Definition. Let $\pi$ be a projective plane. $\pi$ is said to be $(m,\mathrm{\ell })$-transitive if there are lines $m,\mathrm{\ell }$ such that $\pi$ is $(C,\mathrm{\ell })$-transitive for all $C\in m$.

Definition. A projective plane $\pi$ is a translation plane if there is a line $\mathrm{\ell }$ such that $\pi$ is $(\mathrm{\ell },\mathrm{\ell })$-transitive. We also say that $\pi$ is a translation plane with respect to $\mathrm{\ell }$. The line $\mathrm{\ell }$ is called a translation line of $\pi$.

It can be shown that $\pi$ is a translation plane with respect to $\mathrm{\ell }$ iff it can be coordinatized by a Veblen-Wedderburn system (thus implying that $(R,+)$ is abelian).

When $\pi$ is a translation plane with respect to two distinct lines $\mathrm{\ell }$ and $m$, then it is not hard to see that it is a translation plane with respect to every line passing through $\mathrm{\ell }\cap m$.

When $\pi$ is a translation plane with respect to three non-concurrent lines, then it is a translation plane with respect to every line. A projective plane which is a translation plane with respect to every line is called a Moufang plane. An example of a translation plane that is not Moufang is the Hall plane, coordinatized by the Hall quasifield. An example of a projective plane that is not a translation plane is the Hughes plane.

Remark. There are also duals to the notions above: a projective plane $\pi$ is

1. $\mathrm{(}P\mathrm{,}Q\mathrm{)}$-transitive if there are points $P,Q$ such that $\pi$ is $(P,m)$-transitive for any line $m$ passing through $Q$.

2. a dual translation plane if there is a point $P$ such that $\pi$ is $(P,P)$-transitive. We also say that $\pi$ is a dual translation plane with respect to $P$, and that $P$ is a translation point of $\pi$.

If $\pi$ is a projective plane, then the following are true:

• $\pi$ is translation plane with respect to some line $\mathrm{\ell }$ and a dual translation plane with respect to some $P\in \mathrm{\ell }$ iff $\pi$ can be coordinatized by a semifield. In this coordinatization, $\mathrm{\ell }$ is the line at infinity and $P$ is the point with coordinate $(\mathrm{\infty })$.

• $\pi$ is translation plane with respect to some line $PQ$ and $(P,Q)$- and $(Q,P)$-transitive iff $\pi$ can be coordinatized by a nearfield. In this coordinatization, $PQ$ is the line at infinity where $P$ and $Q$ have coordinates $(0)$ and $(\mathrm{\infty })$ (or vice versa).

Remark. By removing the line at infinity from a translation plane, we obtain an affine translation plane. By the definition of a translation plane, an affine translation plane can be characterized as an affine plane where the minor affine Desarguesian property holds.