Ε-quadratic form

In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; $$epsilon = pm 1, accordingly for symmetric or skew-symmetric. They are also called $$(-)^n-quadratic forms, particularly in the context of surgery theory.

There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms, Hermitian forms, and skew-Hermitian forms.

The theory is 2-local: away from 2, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism.

Definition

ε-symmetric forms and ε-quadratic forms are defined thus [ [http://arxiv.org/pdf/math/0111315 Foundations of algebraic surgery] , by Andrew Ranicki, p. 6] .

Given a module $$M over a *-ring $$R, let $$B(M) be the space of bilinear forms on $$M, and let $$Tcolon B(M) o B(M) be the "conjugate transpose" involution $$B(u,v) mapsto B(v,u)^*. Let $$epsilon=pm 1; then $$epsilon T is also an involution. Define the ε-symmetric forms as the invariants of $$epsilon T, and the ε-quadratic forms are the coinvariants.

As a short exact sequence,:$$Q^epsilon o B(M) stackrel{1-epsilon T}{longrightarrow} B(M) o Q_epsilonAs kernel (algebra) and cokernel,:$$Q^epsilon := mbox{ker},(1-epsilon T):$$Q_epsilon := mbox{coker},(1-epsilon T)

The notation $$Q^epsilon, Q_epsilon follows the standard notation $$M^G, M_G for the invariants and coinvariants for a group action, here of the order 2 group (an involution).

Generalization from *

If the * is trivial, then $$epsilon=pm 1, and "away from 2" means that 2 is invertible: $$frac{1}{2} in R.

More generally, one can take for $$epsilon in R any element such that $$epsilon^*epsilon=1.$$epsilon=pm 1 always satisfy this, but so does any element of norm 1, such as complex numbers of unit norm.

Similarly, in the presence of a non-trivial *, ε-symmetric forms are equivalent ε-quadratic forms if there is an element $$lambda in R such that $$lambda^* + lambda = 1. If * is trivial, this is equivalent to $$2lambda=1 or $$lambda = frac{1}{2}.

For instance, in the ring $$R=mathbf{Z}left [ extstyle{frac{1+i}{2 ight] (the integral lattice for the quadratic form $$2x^2-2x+1), with complex conjugation, $$lambda= extstyle{frac{1+i}{2 is such an element, though $$frac{1}{2} otin R.

Intuition

In terms of matrices, (we take $$V to be 2-dimensional):
* matrices $$egin{pmatrix}a & b\c & dend{pmatrix} correspond to bilinear forms
* the subspace of symmetric matrices $$egin{pmatrix}a & b\b & cend{pmatrix} correspond to symmetric forms
* the bilinear form $$egin{pmatrix}a & b\c & dend{pmatrix} yields the quadratic form $$ax^2 + bxy+cyx + dy^2 = ax^2 + (b+c)xy + dy^2 , which is a quotient map with kernel $$egin{pmatrix}0 & b\-b & 0end{pmatrix}.

Refinements

An intuitive way to understand an ε-quadratic form is to think of it as a quadratic refinement of its associated ε-symmetric form.

For instance, in defining a Clifford algebra over a general field or ring, one quotients the tensor algebra by relations coming from the symmetric form "and" the quadratic form: $$vw+wv=2B(v,w) and $$v^2=Q(v).If 2 is invertible, this second relation follows from the first (as the quadratic form can be recovered from the associated bilinear form), but at 2 this additional refinement is necessary.

Example

Given an oriented surface $$Sigma embedded in $$mathbf{R}^3, the middle homology group $$H_1(Sigma) carries not only a skew-symmetric form (via intersection), but also a skew-quadratic form, which can be seen as a quadratic refinement, via self-linking.

For the standard embedded torus, the skew-symmetric form is given by $$egin{pmatrix}0 & 1\-1 & 0end{pmatrix} (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by $$xy with respect to this basis: $$Q(1,0)=Q(0,1)=0: the basis curves don't self-link; and $$Q(1,1)=1: a (1,1) self-links, as in the Hopf fibration.(This form has Arf invariant 0.)

Applications

A key application is in algebraic surgery theory, where even L-groups are defined as Witt groups of ε-quadratic forms, by Andrew Ranicki.Mischenko had previously used ε-symmetric forms, which are not the correct theory for application to surgery.

References