Klein quadric

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 T the points that represent a line of 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 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 p₁₂p₃₄+p₁₃p₄₂+p₁₄p₂₃ = 0 defining Q,where p_ij = u_iv_j-u_jv_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 $C\text{'}$. 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

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