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;, accordingly for symmetric or skew-symmetric. They are also called -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. More briefly, one may refer to quadratic, skew-quadratic, symmetric, and skew-symmetric forms, where "skew" means and the * is implied.
The theory is 2-local: away from 2, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map gives an explicit isomorphism.
Definition
ε-symmetric forms and ε-quadratic forms are defined as follows.Given a module M over a *-ring R, let B be the space of bilinear forms on M, and let be the "conjugate transpose" involution. Since multiplication by −1 is also an involution and commutes with linear maps, −T is also an involution. Thus we can write and εT is an involution, either T or −T. Define the ε-symmetric forms as the invariants of εT, and the ε-quadratic forms are the coinvariants.
As an exact sequence,
As kernel and cokernel,
The notation Qε, Qε follows the standard notation MG, MG for the invariants and coinvariants for a group action, here of the order 2 group.
Composition of the inclusion and quotient maps as yields a map Qε → Qε: every ε-symmetric form determines an ε-quadratic form.
Symmetrization
Conversely, one can define a reverse homomorphism, called the symmetrization map by taking any lift of a quadratic form and multiplying it by. This is a symmetric form because, so it is in the kernel. More precisely,. The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of, but this vanishes after multiplying by. Thus every ε-quadratic form determines an ε-symmetric form.Composing these two maps either way: or yields multiplication by 2, and thus these maps are bijective if 2 is invertible in R, with the inverse given by multiplication with 1/2.
An ε-quadratic form is called non-degenerate if the associated ε-symmetric form is non-degenerate.
Generalization from *
If the * is trivial, then, and "away from 2" means that 2 is invertible:.More generally, one can take for any element such that. 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 to ε-quadratic forms if there is an element such that. If * is trivial, this is equivalent to or, while if * is non-trivial there can be multiple possible λ; for example, over the complex numbers any number with real part 1/2 is such a λ.
For instance, in the ring , with complex conjugation, are two such elements, though.
Intuition
In terms of matrices, if * is trivial:- matrices correspond to bilinear forms
- the subspace of symmetric matrices correspond to symmetric forms
- the subspace of -symmetric matrices correspond to symplectic forms
- the bilinear form yields the quadratic form
- the map 1 + T from quadratic forms to symmetric forms maps
If is complex conjugation, then
- the subspace of symmetric matrices are the Hermitian matrices
- the subspace of skew-symmetric matrices are the skew-Hermitian matrices
Refinements
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: and. If 2 is invertible, this second relation follows from the first, but at 2 this additional refinement is necessary.
Examples
An easy example for an ε-quadratic form is the standard hyperbolic ε-quadratic form. It is given by the bilinear form. The standard hyperbolic ε-quadratic form is needed for the definition of L-theory.For the field of two elements there is no difference between -quadratic and -quadratic forms, which are just called quadratic forms. The Arf invariant of a nonsingular quadratic form over F2 is an F2-valued invariant with important applications in both algebra and topology, and plays a role similar to that played by the discriminant of a quadratic form in characteristic not equal to two.
Manifolds
The free part of the middle homology group of an oriented even-dimensional manifold has an ε-symmetric form, via Poincaré duality, the intersection form. In the case of singly even dimension, this is skew-symmetric, while for doubly even dimension 4k, this is symmetric. Geometrically this corresponds to intersection, where two n/2-dimensional submanifolds in an n-dimensional manifold generically intersect in a 0-dimensional submanifold, by adding codimension. For singly even dimension the order switches sign, while for doubly even dimension order does not change sign, hence the ε-symmetry. The simplest cases are for the product of spheres, where the product and respectively give the symmetric form and skew-symmetric form In dimension two, this yields a torus, and taking the connected sum of g tori yields the surface of genus g, whose middle homology has the standard hyperbolic form.With additional structure, this ε-symmetric form can be refined to an ε-quadratic form. For doubly even dimension this is integer valued, while for singly even dimension this is only defined up to parity, and takes values in Z/2. For example, given a framed manifold, one can produce such a refinement. For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant.
Given an oriented surface Σ embedded in R3, the middle homology group H1 carries not only a skew-symmetric form, but also a skew-quadratic form, which can be seen as a quadratic refinement, via self-linking. The skew-symmetric form is an invariant of the surface Σ, whereas the skew-quadratic form is an invariant of the embedding, e.g. for the Seifert surface of a knot. The Arf invariant of the skew-quadratic form is a framed cobordism invariant generating the first stable homotopy group.
For the standard embedded torus, the skew-symmetric form is given by , and the skew-quadratic refinement is given by xy with respect to this basis: : the basis curves don't self-link; and : a self-links, as in the Hopf fibration.