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; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called ( − ) n {\displaystyle (-)^{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 (= symplectic 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 * (involution) is implied. 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 as follows. Given a module M over a *-ring R, let B(M) be the space of bilinear forms on M, and let T : B(M) → B(M) be the "conjugate transpose" involution B(u, v) ↦ B(v, u)*. Since multiplication by −1 is also an involution and commutes with linear maps, −T is also an involution. Thus we can write ε = ±1 and εT is an involution, either T or −T (ε can be more general than ±1; see below). Define the ε-symmetric forms as the invariants of εT, and the ε-quadratic forms are the coinvariants. As an exact sequence,
0 → Q ε ( M ) → B ( M ) ⟶ 1 − ε T B ( M ) → Q ε ( M ) → 0 {\displaystyle 0\to Q^{\varepsilon }(M)\to B(M){\stackrel {1-\varepsilon T}{\longrightarrow }}B(M)\to Q_{\varepsilon }(M)\to 0}
As kernel and cokernel,
Q ε ( M ) := ker ( 1 − ε T ) {\displaystyle Q^{\varepsilon }(M):={\mbox{ker}}\,(1-\varepsilon T)}
Q ε ( M ) := coker ( 1 − ε T ) {\displaystyle Q_{\varepsilon }(M):={\mbox{coker}}\,(1-\varepsilon T)}
The notation Qε(M), Qε(M) follows the standard notation MG, MG for the invariants and coinvariants for a group action, here of the order 2 group (an involution). Composition of the inclusion and quotient maps (but not 1 − εT) as Q ε ( M ) → B ( M ) → Q ε ( M ) {\displaystyle Q^{\varepsilon }(M)\to B(M)\to Q_{\varepsilon }(M)} yields a map Qε(M) → Qε(M): every ε-symmetric form determines an ε-quadratic form.
Symmetrization Conversely, one can define a reverse homomorphism "1 + εT": Qε(M) → Qε(M), called the symmetrization map (since it yields a symmetric form) by taking any lift of a quadratic form and multiplying it by 1 + εT. This is a symmetric form because (1 − εT)(1 + εT) = 1 − T2 = 0, so it is in the kernel. More precisely, ( 1 + ε T ) B ( M ) < Q ε ( M ) {\displaystyle (1+\varepsilon T)B(M)<Q^{\varepsilon }(M)} . The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of (1 − εT), but this vanishes after multiplying by 1 + εT. Thus every ε-quadratic form determines an ε-symmetric form. Composing these two maps either way: Qε(M) → Qε(M) → Qε(M) or Qε(M) → Qε(M) → Qε(M) 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 ψ ∈ Qε(M) is called non-degenerate if the associated ε-symmetric form (1 + εT)(ψ) is non-degenerate.
Generalization from * If the * is trivial, then ε = ±1, and "away from 2" means that 2 is invertible: 1/2 ∈ R. More generally, one can take for ε ∈ R any element such that ε*ε = 1. ε = ±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 to ε-quadratic forms if there is an element λ ∈ R such that λ* + λ = 1. If * is trivial, this is equivalent to 2λ = 1 or λ = 1/2, 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 R = Z [ 1 + i 2 ] {\displaystyle R=\mathbf {Z} \left[\textstyle {\frac {1+i}{2}}\right]} (the integral lattice for the quadratic form 2x2 − 2x + 1), with complex conjugation, λ = 1 ± i 2 {\displaystyle \lambda =\textstyle {\frac {1\pm i}{2}}} are two such elements, though 1/2 ∉ R.
Intuition In terms of matrices (we take V to be 2-dimensional), if * is trivial:
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