nLab
quadratic form

Definition

For $V$ a vector space or $k$-module, a quadratic form on $V$ is a function

$$q:V\to k$$q\colon V \to k

such that for all $v\in V$, $t\in k$

$$q(tv)=t^{2}q(v)$$q(t v) = t^2 q(v)

and the polarization of $q$

$$(v,w)\mapsto q(v+w)-q(v)-q(w)$$(v,w) \mapsto q(v+w) - q(v) - q(w)

is a bilinear form.

Quadratic refinement

Let

$$⟨-,-⟩:V\otimes V\to k$$\langle -,-\rangle : V \otimes V \to k

be a bilinear form. We say a function $q:V\to k$ is a quadratic refinement of $⟨-,-⟩$ if

$$⟨v,w⟩=q(v+w)-q(v)-q(w)$$\langle v,w\rangle = q(v + w) - q(v) - q(w)

for all $v,w\in V$.

If this exists, then

$$⟨v,v⟩=2q(v).$$\langle v , v \rangle = 2 q(v) \,.

So a quadratic refinement always exists when $2\in k$ is invertible. Otherwise its existence is a non-trivial condition.

References

Course notes include for instance

• On the relation between quadratic and bilinear forms (pdf)

• Bilinear and quadratic forms (pdf)

Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory