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} $).

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)