Moufang plane

Moufang plane

A

projective plane

is called a

Moufang plane

if each of its lines is a

translation line

(it is a translation plane with respect to any line in the plane). In other words, a Moufang plane is a projective plane such that the

minor Desarguesian

property

holds everywhere.

It can be shown that a projective plane is Moufang iff it can be coordinatized by an alternative division ring, which is a

non-associative

division ring satisfying the left and right alternative laws:

(

a

⁢

a

)

⁢

b

=

a

⁢

(

a

⁢

b

)

and

a

⁢

(

b

⁢

b

)

=

(

a

⁢

b

)

⁢

b

,

or, equivalently, a

Veblen-Wedderburn system

where the two alternative laws hold.

For example, any field plane (a projective plane coordinatized by a field) is Moufang, and more generally any skew field plane (such as the quaternion plane

P

⁢

(

2

,

ℍ

)

, as the quaternions form a division ring). However, all of these examples are Desarguesian. An example of a non-Desarguesian Moufang plane that is the

octonion

plane,

P

⁢

(

2

,

𝕆

)

(where the

coordinates

are octonions), for the

multiplication

on the octonions are alternative and not associative. It is interesting to note that, while one can construct higher dimensional

projective spaces

over any division ring, no such construction is possible if the division ring is non-associative, as any coordinatization of a projective space of

dimension

greater than

2

is always a division ring. An example of a translation plane that is not Moufang is the Hall plane.

According to the

definition

above, a Moufang plane is a translation plane in which every line is a translation line. So are there any planes intermediate between a translation plane and a Moufang plane, in the sense that a translation plane with exactly two, three, more more translation lines? It turns out that, if a translation plane has two distinct translation lines, say

ℓ

1

,

ℓ

2

, then every line

passing through

ℓ

1

∩

ℓ

2

is also a translation line, and, as a result, every line of the plane is a translation line, which means the plane itself is Moufang.

Again, by the definition, it is not hard to see that any

ternary ring

coordinatizing a Moufang plane

π

is an alternative division ring. In fact, it can be shown that any two alternative division rings coordinatizing

π

are

isomorphic

. This is the result of an interesting

algebraic

fact by Bruck and Kleinfeld: any alternative division ring is either an associative division ring or a

Cayley algebra

over its center (which at the same time is a field).

Another algebraic result that has an interesting geometric consequence is the Artin-Zorn’s

theorem

, which asserts that a finite alternative division ring is a field. This implies that a finite Moufang plane is a field plane!

References

1

M. Hall, Jr.,

The Theory of Groups

, Macmillan (1959)

2

R. Artzy,

Linear

Geometry

, Addison-Wesley (1965)

3

R. H. Bruck, E. Kleinfeld,

The Structure of Alternative Division Rings

, Proc. Amer. Math. Soc. 78, pp.464-481 (1955)

Title

Moufang plane

Canonical

name

MoufangPlane

Date of creation

2013-03-22 19:15:27

Last modified on

2013-03-22 19:15:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A40

Classification

msc 51A35

Related topic

TranslationPlane

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