ε-quadratic form

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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.

1 Definition
- 1.1 Generalization from *
2 Intuition
- 2.1 Refinements
3 Example
4 Applications
5 References

[edit] Definition

ε-symmetric forms and ε-quadratic forms are defined thus^[1].

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

As a short exact sequence,

$Q^\epsilon \to B(M) \stackrel{1-\epsilon T}{\longrightarrow} B(M) \to Q_\epsilon$

As kernel (algebra) and cokernel,

$Q^\epsilon := \mbox{ker}\,(1-\epsilon T)$

$Q_\epsilon := \mbox{coker}\,(1-\epsilon T)$

The notation $Q ε, Q ε$ 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).

[edit] 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 $ε * ε = 1$ . $\epsilon=\pm 1$ 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 $λ * + λ = 1$ . If * is trivial, this is equivalent to $2λ = 1$ or $\lambda = \frac{1}{2}$ .

For instance, in the ring $R=\mathbf{Z}\left[\textstyle{\frac{1+i}{2}}\right]$ (the integral lattice for the quadratic form $2 x 2 - 2 x + 1$ ), with complex conjugation, $\lambda=\textstyle{\frac{1+i}{2}}$ is such an element, though $\frac{1}{2} \not\in R$ .

[edit] Intuition

In terms of matrices, (we take $V$ to be 2-dimensional):

matrices $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ correspond to bilinear forms
the subspace of symmetric matrices $\begin{pmatrix}a & b\\b & c\end{pmatrix}$ correspond to symmetric forms
the bilinear form $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ yields the quadratic form $a x 2 + b x y + c y x + d y 2 = a x 2 + (b + c) x y + d y 2$ , which is a quotient map with kernel $\begin{pmatrix}0 & b\\-b & 0\end{pmatrix}$ .

[edit] 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: $v w + w v = 2 B (v, w)$ 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.

[edit] Example

Given an oriented surface $Σ$ embedded in $\mathbf{R}^3$ , the middle homology group $H 1 (Σ)$ 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 $\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$ (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by $x y$ 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.)

[edit] 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.