quadratic map

Given a commutative ring K and two K-modules M and N then a map q:MN is called quadratic if

  1. 1.

    q(αx)=α2q(x) for all xM and αK.

  2. 2.

    b(x,y):=q(x+y)-q(x)-q(y), for x,yM, is a bilinear map.

The only differencePlanetmathPlanetmath between quadratic maps and quadratic formsMathworldPlanetmath is the insistence on the codomain N instead of a K. So in this way every quadratic form is a special case of a quadratic map. Most of the properties for quadratic forms apply to quadratic maps as well. For instance, if K has no 2-torsion (2x=0 implies x=0) then


defines a symmetricPlanetmathPlanetmathPlanetmathPlanetmath K-bilinear map c:M×MN with c(x,x)=q(x). In particular if 1/2K then c(x,y)=12b(x,y). This definition is one instance of a polarization (i.e.: substituting a single variable in a formulaMathworldPlanetmathPlanetmath with x+y and comparing the result with the formula over x and y separately.) Continuing without 2-torsionPlanetmathPlanetmath, if b is a symmetric K-bilinear map (perhaps not a form) then defining qb(x)=b(x,x) determines a quadratic map since




Have have no 2-torsion we can recover b form qb. So in odd and 0 characteristic rings we find symmetric bilinear maps and quadratic maps are in 1-1 correspondence.

An alternative understanding of b is to treat this as the obstruction to q being an additivePlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Thus a submodule T of M for which b(T,T)=0 is a submodule of M on which q|T is an additive homomorphism. Of course because of the first condition, q is semi-linear on T only when αα2 is an automorphismPlanetmathPlanetmath of K, in particular, if K has characteristic 2. When the characteristic of K is odd or 0 then q(T)=0 if and only if b(T,T)=0 simply because q(x)=b(x,x) (or up to a 1/2 multipleMathworldPlanetmathPlanetmath depending on conventions). However, in characteristic 2 it is possible for b(T,T)=0 yet q(T)0. For instance, we can have q(x)0 yet b(x,x)=q(2x)-q(x)-q(x)=0. This is summed up in the following definition:

A subspacePlanetmathPlanetmathPlanetmath T of M is called totally singular if q(T)=0 and totally isotropic if b(T,T)=0. In odd or 0 characteristic, totally singular subspaces are precisely totally isotropic subspaces.

Title quadratic map
Canonical name QuadraticMap
Date of creation 2013-03-22 16:27:55
Last modified on 2013-03-22 16:27:55
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 9
Author Algeboy (12884)
Entry type DerivationPlanetmathPlanetmath
Classification msc 11E08
Classification msc 11E04
Classification msc 15A63
Related topic QuadraticJordanAlgebra
Related topic IsotropicQuadraticSpace
Defines quadratic map
Defines totally singular
Defines totally isotropic
Defines polarization formula
Defines polarization identity