Normal quadratic subspace

Let $(𝑉, 𝑞)$ be a quadratic space. A normal subspace¹ $𝑈 ⊴ 𝑉$ is a vector subspace consisting of only degenerate isotropic vectors, #m/def/linalg/q i.e. a totally isotropic subspace of the radical $r a d V$ .

Away from 2 a subspace is normal iff it is radical

Clearly a normal subspace must be radical. Let $𝑈 \leq r a d 𝑉$ . Then $𝑈$ is totally isotropic, since for any $𝑢 \in 𝑈$ we have

𝑞 (𝑢) = 12 𝑏 𝑞 (𝑢, 𝑢) = 0

Normal subspaces of $𝑉$ lie in correspondence with congruence relations of $𝑉$ , 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