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.

For the Klein quartic, see Klein quartic.

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 ${\displaystyle C}$ and ${\displaystyle C'}$. 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.

References

• Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry : from foundations to applications, page 169, Cambridge University Press ISBN 978-0521482776
• Arthur Cayley (1873) "On the superlines of a quadric surface in five-dimensional space", Collected Mathematical Papers 9: 79–83.
• Felix Klein (1870) "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen 2: 198
• Oswald Veblen & John Wesley Young (1910) Projective Geometry, volume 1, Interpretation of line coordinates as point coordinates in S₅, page 331, Ginn and Company.
• Ward, Richard Samuel; Wells, Raymond O'Neil, Jr. (1991), Twistor Geometry and Field Theory, Cambridge University Press, ISBN 978-0-521-42268-0.