# Correlation

duality

A bijective mapping $\kappa$ between projective spaces of the same finite dimension such that $S _ {p} \subset S _ {q}$ implies $\kappa ( S _ {q} ) \subset \kappa ( S _ {p} )$. 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 $\Pi _ {n} ( K)$ over a division ring $K$ onto a space $\Pi _ {n} ( L)$ over a division ring $L$ is that there exists an anti-isomorphism $\alpha : K \rightarrow L$, i.e. a bijective mapping for which $\alpha ( x + y ) = \alpha ( x) + \alpha ( y)$, $\alpha ( x y ) = \alpha ( y) \alpha ( x)$; in that case $\Pi _ {n} ( L)$ is dual to $\Pi _ {n} ( K)$. Examples of spaces with an auto-correlation, i.e. a correlation onto itself, are the real projective spaces $( K = \mathbf R , \alpha = \mathop{\rm id} )$, the complex projective spaces $( K = \mathbf C , \alpha : z \rightarrow \overline{z}\; )$ and the quaternion projective spaces $( K = \mathbf H , \alpha : z \rightarrow \overline{z}\; )$.

A polarity is an auto-correlation $\kappa$ satisfying $\kappa ^ {2} = \mathop{\rm id}$. A projective space $\Pi _ {n} ( K)$ over a division ring $K$ admits a polarity if and only if $K$ admits an involutory anti-automorphism, i.e. an anti-automorphism $\alpha$ with $\alpha ^ {2} = \mathop{\rm id}$.

A subspace $W$ is called a null subspace relative to an auto-correlation $\kappa$ if $P \subset \kappa ( P)$ for any point $P \in W$, and strictly isotropic if $W \subset \kappa ( W)$. 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 $\Pi _ {n} ( K)$ over a division ring $K$ be interpreted as the set of linear subspaces of the (left) linear space $K ^ {n+} 1$ over $K$. A semi-bilinear form on $K ^ {n+} 1$ is a mapping $f : K ^ {n+} 1 \times K ^ {n+} 1 \rightarrow K$ together with an anti-automorphism $\alpha$ of $K$ such that

$$f ( x + y , z ) = \ f ( x , z ) + f ( y , z ) ,$$

$$f ( x , y + z ) = f ( x , y ) + f ( x , z ) ,$$

$$f ( k x , y ) = k f ( x , y ) ,$$

$$f ( x , k y ) = f ( x , y ) \alpha ( k) .$$

In particular, if $K$ is a field and $\alpha = \mathop{\rm id}$, then $f$ is a bilinear form. A semi-bilinear form $f$ is called non-degenerate provided $f ( x , y ) = 0$ for all $x$( all $y$) implies $y = 0$( $x = 0$, respectively). Any auto-correlation $\kappa$ of $\Pi _ {n} ( K)$ can be represented with the aid of a non-degenerate semi-bilinear form $f$ in the following way: for a subspace $V$ of $K ^ {n+} 1$ its image is the orthogonal complement of $V$ with respect to $f$:

$$\kappa ( V) = \{ {y \in K ^ {n+} 1 } : {f ( x , y ) = 0 \textrm{ for all } \ x \in V } \}$$

(the Birkhoff–von Neumann theorem, ). $\kappa$ is a polarity if and only if $f$ is reflexive, i.e. if $f ( x , y ) = 0$ implies $f ( y , x ) = 0$. By multiplying $f$ by a suitable element of $K$ one can bring any reflexive non-degenerate semi-bilinear form $f$ and the corresponding automorphism $\alpha$ in either of the following two forms:

1) $\alpha$ is an involution, i.e. $\alpha ^ {2} = \mathop{\rm id}$, and

$$f ( y , x ) = \alpha ( f ( x , y ) ) .$$

In this case one calls $f$ symmetric if $\alpha = \mathop{\rm id}$( and hence necessarily $K$ is a field) and Hermitian if $\alpha \neq \mathop{\rm id}$.

2) $\alpha = \mathop{\rm id}$( and hence $K$ is a field) and

$$f ( y , x ) = - f ( x , y ) .$$

Such an $f$ is called anti-symmetric.

A special example of a correlation is the following. Let $\Pi _ {n} ( K)$ be a projective space over a division ring $K$. Define the opposite division ring $K ^ {o}$ as the set of elements of $K$ with the same addition but with multiplication

$$x \cdot y = \ y x .$$

$\alpha : x \rightarrow x$ is an anti-isomorphism from $K$ onto $K ^ {o}$ which defines the canonical correlation from $\Pi _ {n} ( K)$ onto $\Pi _ {n} ( K ^ {o} )$. The (left) projective space $\Pi _ {n} ( K ^ {o} )$, which can be identified with the right projective space $\Pi _ {n} ( K) ^ {*}$, i.e. with the set of linear subspaces of the $( n + 1 )$- dimensional right vector space $K ^ {n+} 1$, is the (canonical) dual space of $\Pi _ {n} ( K)$( cf. Projective algebra, the construction of $\Pi _ {n}$).