Klein quadric explained

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

p _{12} p _{34} + p _{13} p _{42} + p _{14} p _{23} = 0 defining Q, where 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.

References

• • Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry : from foundations to applications, page 169, Cambridge University Press 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.
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