Polarity

polar transformation

A correlation for which , that is, if and only if . A polarity divides all subspaces into pairs; in particular, if a pair is formed by the subspaces and , where is a point and is a hyperplane, then is called the pole of the hyperplane and is called the polar of the point . A projective space over the skew-field has a polarity if and only if admits an involutory anti-automorphism (that is, ). Suppose that is represented by a semi-bilinear form . Then is a polarity if and only if implies .

A polarity is either a symplectic correlation, characterized by the fact that for every point (in this case, is a skew-symmetric form on , while is a field), or can be represented as an -symmetric form on : (a symmetric polarity), in this case the existence of a non-strictly isotropic null subspace is equivalent to the fact that the characteristic of the skew-field equals 2 (in particular, if , then any null subspace is strictly isotropic).

Relative to a polarity one defines decomposition of a projective space into subspaces, which makes it possible to reduce the semi-bilinear form representing to canonical form. The most important among these are the following:

— a maximal non-isotropic null subspace; its dimension is , where is even and is called the deficiency of , and is skew-symmetric;

— a maximal strictly-isotropic subspace; its dimension is , is called the index, ;

— a component, free or null subspace, non-isotropic, where is positive or negative definite, .

— a maximal null subspace; its dimension is .

A projective transformation is called -admissible (relative to the polarity ) if . A semi-linear transformation induces a -admissible projective transformation if and only if in there is a for which . The -admissible transformations form a group, (called the polarity group). If the group is transitive, either every point of the space is null (and is called symplectic) or there is no null point (and in this case is called orthogonal for , and unitary for ).

References

[1]	N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian)

Comments

Let be a bipartite graph, and let be the corresponding partition of . A polarity on is an automorphism of the graph such that and , .

The term polarity is mostly encountered in a geometric setting such as that of a projective space or incidence system. In this case the two sets of vertices are the lines and points of the incidence structure and there is an edge between a "point-vertex" and a "line-vertex" if and only if the point and line are incident.

The classical setting is that of a projective space with a non-degenerate bilinear form on it. The corresponding polarity between -dimensional subspaces and -dimensional subspaces is defined by .

In the setting of a (Desarguesian or not) projective space a polarity is also viewed as a symmetric relation such that for all , is either a hyperplane or itself. If , the polarity is non-degenerate. A subspace is totally isotropic if .

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2]	H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)
[a3]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a4]	R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952)
[a5]	D. Pedoe, "Geometry. A comprehensive course" , Dover, reprint (1988) pp. Sect. 85.5
[a6]	P. Dembowsky, "Finite geometries" , Springer (1968)