Normal quadratic subspace

Let be a quadratic space. A normal subspace¹ is a vector subspace consisting of only degenerate isotropic vectors, #m/def/linalg/q i.e. a totally isotropic subspace of the radical .

Away from 2 a subspace is normal iff it is radical

Clearly a normal subspace must be radical. Let . Then is totally isotropic, since for any we have

Normal subspaces of lie in correspondence with congruence relations of , hence they may be used to form the Quotient quadratic space.

#state/tidy | #lang/en | #SemBr

This terminology is nonstandard, but nice. As Jeff Saunders remarks, it is not only reminiscent of Normal subgroup, but also the fact that such a subspace is “normal” to everything else, under the bilinear form. ↩

Footnotes