Klein quadric

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For the Klein quartic, see Klein quartic.

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 hyperbolic 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

defining Q, where

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 and . 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.