Klein quadric

In mathematics, the lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a quadric, Q known as the Klein quadric.

If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ²V of V. The line coordinates obtained this way are known as Plücker coordinates.

These Plücker coordinates satisfy the quadratic relation

${\displaystyle p_{12}p_{34}+p_{13}p_{42}+p_{14}p_{23}=0}$

defining Q, where

${\displaystyle p_{ij}=u_{i}v_{j}-u_{j}v_{i}}$

are the coordinates of the line spanned by the two vectors u and v.

The 3-space, S, can be reconstructed again from the quadric, Q: the planes contained in Q fall into two equivalence classes, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be C and C′. The geometry of S is retrieved as follows:

1. The points of S are the planes in C.
2. The lines of S are the points of Q.
3. The planes of S are the planes in C′.

The fact that the geometries of S and Q are isomorphic can be explained by the isomorphism of the Dynkin diagrams A₃ and D₃.