Quadratic space

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, q i.e. a ^totallyIsotropic subspace of the radical radV.

Away from 2 a subspace is normal iff it is radical

Clearly a normal subspace must be radical. Let 𝑈 ≤rad⁡𝑉. Then 𝑈 is totally isotropic, since for any 𝑢 ∈𝑈 we have

𝑞(𝑢)=12𝑏𝑞(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.

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Footnotes

1. 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. ↩