Correlation

duality

A bijective mapping between projective spaces of the same finite dimension such that implies . The image of a sum of subspaces under a correlation is the intersection of their images and, conversely, the image of an intersection is the sum of the images. In particular, the image of a point is a hyperplane and vice versa. A necessary and sufficient condition for the existence of a correlation of a projective space over a division ring onto a space over a division ring is that there exists an anti-isomorphism , i.e. a bijective mapping for which , ; in that case is dual to . Examples of spaces with an auto-correlation, i.e. a correlation onto itself, are the real projective spaces , the complex projective spaces and the quaternion projective spaces .

A polarity is an auto-correlation satisfying . A projective space over a division ring admits a polarity if and only if admits an involutory anti-automorphism, i.e. an anti-automorphism with .

A subspace is called a null subspace relative to an auto-correlation if for any point , and strictly isotropic if . Any strictly isotropic subspace is a null subspace. A polarity relative to which the whole space is a null space is called a null (or symplectic) polarity (see also Polarity).

Let the projective space over a division ring be interpreted as the set of linear subspaces of the (left) linear space over . A semi-bilinear form on is a mapping together with an anti-automorphism of such that

In particular, if is a field and , then is a bilinear form. A semi-bilinear form is called non-degenerate provided for all (all ) implies (, respectively). Any auto-correlation of can be represented with the aid of a non-degenerate semi-bilinear form in the following way: for a subspace of its image is the orthogonal complement of with respect to :

(the Birkhoff–von Neumann theorem, ). is a polarity if and only if is reflexive, i.e. if implies . By multiplying by a suitable element of one can bring any reflexive non-degenerate semi-bilinear form and the corresponding automorphism in either of the following two forms:

1) is an involution, i.e. , and

In this case one calls symmetric if (and hence necessarily is a field) and Hermitian if .

2) (and hence is a field) and

Such an is called anti-symmetric.

A special example of a correlation is the following. Let be a projective space over a division ring . Define the opposite division ring as the set of elements of with the same addition but with multiplication

is an anti-isomorphism from onto which defines the canonical correlation from onto . The (left) projective space , which can be identified with the right projective space , i.e. with the set of linear subspaces of the -dimensional right vector space , is the (canonical) dual space of (cf. Projective algebra, the construction of ).

References

[a1]	R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952)
[a2]	G. Birkhoff, J. von Neumann, "The logic of quantum mechanics" Ann. of Math. , 37 (1936) pp. 823–843
[a3]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1963)
[a4]	D.R. Hughes, F.C. Piper, "Projective planes" , Springer (1972)