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

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

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:

The points of S are the planes in C.
The lines of S are the points of Q.
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

Ward, Richard Samuel; Wells, Raymond O'Neil, Jr. (1991), Twistor Geometry and Field Theory, Cambridge University Press, ISBN 978-0-521-42268-0 .

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