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
).
Comments
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) |
Correlation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation&oldid=13946