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,:$Q^epsilon\; o\; B(M)\; stackrel\{1-epsilon\; T\}\{longrightarrow\}\; B(M)\; o\; Q\_epsilon$As 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 correspond to bilinear forms
* the subspace of symmetric matrices correspond to symmetric forms
* the bilinear form yields the quadratic form $ax^2\; +\; bxy+cyx\; +\; dy^2\; =\; ax^2\; +\; (b+c)xy\; +\; dy^2$, which is a quotient map with kernel .

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 (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