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# 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 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 Λ^{2}*V* 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 *A*_{3} and *D*_{3}.

References

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

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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