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quadratic form

Contents

Definition

For VV a vector space or more generally a kk-module, then a quadratic form on VV is a function

q:Vk q\colon V \to k

such that for all vVv \in V, tkt \in k

q(tv)=t 2q(v) q(t v) = t^2 q(v)

and the polarization of qq

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

is a bilinear form.

Quadratic refinement

Let

,:VVk \langle -,-\rangle \colon V \otimes V \to k

be a bilinear form. A function

q:Vk q \colon V \to k

is called a quadratic refinement of ,\langle -,-\rangle if

v,w=q(v+w)q(v)q(w)+q(0) \langle v,w\rangle = q(v + w) - q(v) - q(w) + q(0)

for all v,wVv,w \in V.

If such qq is indeed a quadratic form in that q(tv)=t 2q(v)q(t v) = t^2 q(v) then q(0)=0q(0) = 0 and

v,v=2q(v). \langle v , v \rangle = 2 q(v) \,.

This means that a quadratic refinement by a quadratic form always exists when 2k2 \in k is invertible. Otherwise its existence is a non-trivial condition. One way to express quadratic refinements is by characteristic elements of a bilinear form. See there for more.

References

Course notes include for instance

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

  • Bilinear and quadratic forms (pdf)

  • section 10 in Analytic theory of modular forms (pdf)

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

Revised on August 21, 2014 04:48:49 by Urs Schreiber (193.175.4.219)