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) |
Polarity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polarity&oldid=48230