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 ${\mathrm{\Pi }}_n(K)$ over a division ring $K$ onto a space ${\mathrm{\Pi }}_n(L)$ over a division ring $L$ is that there exists an anti-isomorphism $\alpha :K\to L$, i.e. a bijective mapping for which $\alpha (x+y)=\alpha (x)+\alpha (y)$, $\alpha (xy)=\alpha (y)\alpha (x)$; in that case ${\mathrm{\Pi }}_n(L)$ is dual to ${\mathrm{\Pi }}_n(K)$. Examples of spaces with an auto-correlation, i.e. a correlation onto itself, are the real projective spaces $(K=\mathbf{R},\alpha =\mathrm{id})$, the complex projective spaces $(K=\mathbf{C},\alpha :z\to \bar{z})$ and the quaternion projective spaces $(K=\mathbf{H},\alpha :z\to \bar{z})$.

A polarity is an auto-correlation $\kappa$ satisfying ${\kappa }^2=\mathrm{id}$. A projective space ${\mathrm{\Pi }}_n(K)$ 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 ${\alpha }^2=\mathrm{id}$.

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 ${\mathrm{\Pi }}_n(K)$ 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 $f:K^{n+}1\times K^{n+}1\to K$ together with an anti-automorphism $\alpha$ of $K$ such that

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

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

$$f(kx,y)=kf(x,y),$$

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

In particular, if $K$ is a field and $\alpha =\mathrm{id}$, 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 ${\mathrm{\Pi }}_n(K)$ 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 $f$:

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

(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. ${\alpha }^2=\mathrm{id}$, and

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

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

2) $\alpha =\mathrm{id}$( 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 ${\mathrm{\Pi }}_n(K)$ 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

$$x\cdot y=yx.$$

$\alpha :x\to x$ is an anti-isomorphism from $K$ onto $K^o$ which defines the canonical correlation from ${\mathrm{\Pi }}_n(K)$ onto ${\mathrm{\Pi }}_n(K^o)$. The (left) projective space ${\mathrm{\Pi }}_n(K^o)$, which can be identified with the right projective space ${\mathrm{\Pi }}_n(K{)}^{\ast }$, 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 ${\mathrm{\Pi }}_n(K)$( cf. Projective algebra, the construction of ${\mathrm{\Pi }}_n$).

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