Sesquilinear form

Sesquilinear forms are not restricted to the complex numbers and may be defined on any K-module where K is a division ring (see below).

Informal introduction

Sesquilinear forms abstract and generalize the basic notion of a Hermitian form on complex vector space. Hermitian forms are commonly seen in physics, as the inner product on a complex Hilbert space. In such cases, the standard Hermitian form on Cn is given by :\langle w,z \rangle = \sum_{i=1}^n \overline{w}_i z_i.

where \overline{w}_i denotes the complex conjugate of w_i ~. This product may be generalized to situations where one is not working with an orthonormal basis for Cn, or even any basis at all. By inserting an extra factor of i into the product, one obtains the skew-Hermitian form, defined more precisely, below. There is no particular reason to restrict the definition to the complex numbers; it can be defined for arbitrary rings carrying an antiautomorphism, informally understood to be a generalized concept of "complex conjugation" for the ring.

Convention

Conventions differ as to which argument should be linear. In the commutative case, we shall take the first to be linear, as is common in the mathematical literature, except in the section devoted to sesquilinear forms on complex vector spaces. There we use the other convention and take the first argument to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used mostly by physicists and originates in Dirac's bra–ket notation in quantum mechanics. It is also consistent with the definition of the usual (Euclidean) product of w,z\in\mathbb C^n as w^*z.

In the more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear.

Complex vector spaces

:Assumption: In this section, sesquilinear forms are antilinear in their first argument and linear in their second.

Over a complex vector space V a map \varphi : V \times V \to \Complex is sesquilinear if :\begin{align} &\varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w)\ &\varphi(a x, b y) = \overline{a}b,\varphi(x,y)\end{align} for all x, y, z, w \in V and all a, b \in \Complex. Here, \overline{a} is the complex conjugate of a scalar a.

A complex sesquilinear form can also be viewed as a complex bilinear map \overline{V} \times V \to \Complex where \overline{V} is the complex conjugate vector space to V. By the universal property of tensor products these are in one-to-one correspondence with complex linear maps \overline{V} \otimes V \to \Complex.

For a fixed z \in V the map w \mapsto \varphi(z, w) is a linear functional on V (i.e. an element of the dual space V^*). Likewise, the map w \mapsto \varphi(w, z) is a conjugate-linear functional on V.

Given any complex sesquilinear form \varphi on V we can define a second complex sesquilinear form \psi via the conjugate transpose: \psi(w,z) = \overline{\varphi(z,w)}. In general, \psi and \varphi will be different. If they are the same then \varphi is said to be Hermitian. If they are negatives of one another, then \varphi is said to be skew-Hermitian. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.

Matrix representation

If V is a finite-dimensional complex vector space, then relative to any basis \left{ e_i \right}i of V, a sesquilinear form is represented by a matrix A, and given by \varphi(w,z) = \varphi \left(\sum_i w_i e_i, \sum_j z_j e_j \right) = \sum_i \sum_j \overline{w_i} z_j \varphi\left(e_i, e_j\right) = w^\dagger A z . where w^\dagger is the conjugate transpose. The components of the matrix A are given by A{ij} := \varphi\left(e_i, e_j\right).

Hermitian form

:The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain differential form on a Hermitian manifold.

A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V \times V \to \Complex such that h(w,z) = \overline{h(z, w)}. The standard Hermitian form on \Complex^n is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by \langle w,z \rangle = \sum_{i=1}^n \overline{w}_i z_i. More generally, the inner product on any complex Hilbert space is a Hermitian form.

A minus sign is introduced in the Hermitian form w w^* - z z^* to define the group SU(1,1).

A vector space with a Hermitian form (V, h) is called a Hermitian space.

The matrix representation of a complex Hermitian form is a Hermitian matrix.

A complex Hermitian form applied to a single vector |z|_h = h(z, z) is always a real number. One can show that a complex sesquilinear form is Hermitian if and only if the associated quadratic form is real for all z \in V.

Skew-Hermitian form

A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form s : V \times V \to \Complex such that s(w,z) = -\overline{s(z, w)}. Every complex skew-Hermitian form can be written as the imaginary unit i := \sqrt{-1} times a Hermitian form.

The matrix representation of a complex skew-Hermitian form is a skew-Hermitian matrix.

A complex skew-Hermitian form applied to a single vector |z|_s = s(z, z) is always a purely imaginary number.

Over a division ring

This section applies unchanged when the division ring K is commutative. More specific terminology then also applies: the division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions.

Definition

A σ-sesquilinear form over a right K-module M is a bi-additive map φ : M × M → K with an associated anti-automorphism σ of a division ring K such that, for all x, y in M and all α, β in K, :\varphi(x \alpha, y \beta) = \sigma(\alpha) , \varphi(x, y) , \beta .

The associated anti-automorphism σ for any nonzero sesquilinear form φ is uniquely determined by φ.

Orthogonality

Given a sesquilinear form φ over a module M and a subspace (submodule) W of M, the orthogonal complement of W with respect to φ is :W^{\perp}={\mathbf{v} \in M \mid \varphi (\mathbf{v}, \mathbf{w})=0,\ \forall \mathbf{w}\in W} .

Similarly, x ∈ M is orthogonal to y ∈ M with respect to φ, written x ⊥φ y (or simply x ⊥ y if φ can be inferred from the context), when . This relation need not be symmetric, i.e. x ⊥ y does not imply y ⊥ x (but see ** below).

Reflexivity

A sesquilinear form φ is reflexive if, for all x, y in M, :\varphi(x, y) = 0 implies \varphi(y, x) = 0. That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric.

Hermitian variations

A σ-sesquilinear form φ is called (σ, ε)-Hermitian if there exists ε in K such that, for all x, y in M, :\varphi(x, y) = \sigma ( \varphi (y, x)) , \varepsilon . If , the form is called σ-Hermitian, and if , it is called σ-anti-Hermitian. (When σ is implied, respectively simply Hermitian or anti-Hermitian.)

For a nonzero (σ, ε)-Hermitian form, it follows that for all α in K, : \sigma ( \varepsilon ) = \varepsilon^{-1} : \sigma ( \sigma ( \alpha ) ) = \varepsilon \alpha \varepsilon^{-1} . It also follows that φ(x, x) is a fixed point of the map α ↦ σ(α)ε. The fixed points of this map form a subgroup of the additive group of K.

A (σ, ε)-Hermitian form is reflexive, and every reflexive σ-sesquilinear form is (σ, ε)-Hermitian for some ε. – https://books.google.com/books?id=S9q8uKabV60C&pg=PA456 Sesquilinear form at the Encyclopedia of Mathematics – https://books.google.com/books?id=ScvSCQAAQBAJ&pg=PA28

In the special case that σ is the identity map (i.e., ), K is commutative, φ is a bilinear form and . Then for the bilinear form is called symmetric, and for is called skew-symmetric.

Matrix representation of a Hermitian form

---(as for over a field) Given an ordered basis { e**i } of the vector space V, a sesquilinear form φ on V uniquely determines the matrix M**φ by: : \varphi (x, y) = x M_\varphi \sigma (y)^{\rm T}.

---(needs work) A sesquilinear form can also be viewed as an F-bilinear map : V \times V^* \to F , where V∗ is the dual space of V.

Example

Let V be the three dimensional vector space over the finite field , where q is a prime power. With respect to the standard basis we can write and and define the map φ by: :\varphi(x, y) = x_1 y_1{}^q + x_2 y_2{}^q + x_3 y_3{}^q. The map σ : t ↦ t**q is an involutory automorphism of F. The map φ is then a σ-sesquilinear form. The matrix M**φ associated to this form is the identity matrix. This is a Hermitian form.

In projective geometry

:Assumption: In this section, sesquilinear forms are antilinear (resp. linear) in their second (resp. first) argument.

In a projective geometry G, a permutation δ of the subspaces that inverts inclusion, i.e. : S ⊆ T ⇒ T**δ ⊆ S**δ for all subspaces S, T of G, is called a correlation. A result of Birkhoff and von Neumann (1936) shows that the correlations of desarguesian projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space. A sesquilinear form φ is nondegenerate if for all y in V (if and) only if .

To achieve full generality of this statement, and since every desarguesian projective geometry may be coordinatized by a division ring, Reinhold Baer extended the definition of a sesquilinear form to a division ring, which requires replacing vector spaces by R-modules. (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.) A generalization called a semi-bilinear form was used by Reinhold Baer to characterize linear manifolds (projective spaces) that are dual to each other in chapter 4 of his book Linear Algebra and Projective Geometry (1952). For a vector space A over a skewfield F he requires:

:A pair consisting of an anti-automorphism α of the skewfield F and a function f : A × A → F satisfying :for all a, b, c ∈ A: , , and :for all t ∈ F and x, y ∈ A: , (page 101) :(The "transformation exponential notation" t ↦ tα is adopted in group theory literature.)

Baer calls such a form an α-form over A. The complex sesquilinear form described above has complex conjugation for α. When α is the identity, then f is a bilinear form.

In the algebraic structure called a *-ring the anti-automorphism is denoted by * and forms are constructed as indicated for α. Special constructions such as skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms are all considered in the broader context.

Particularly in L-theory, one also sees the term ε-symmetric form, where , to refer to both symmetric and skew-symmetric forms.

Over arbitrary rings

The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to the nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the arbitrary field version of the definition to arbitrary rings.

Let R be a ring, V an R-module and σ an antiautomorphism of R.

A map φ : V × V → R is σ-sesquilinear if :\varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w) :\varphi(c x, d y) = c , \varphi(x,y) , \sigma(d) for all x, y, z, w in V and all c, d in R.

An element x is orthogonal to another element y with respect to the sesquilinear form φ (written x ⊥ y) if . This relation need not be symmetric, i.e. x ⊥ y does not imply y ⊥ x.

A sesquilinear form φ : V × V → R is reflexive (or orthosymmetric) if implies for all x, y in V.

A sesquilinear form φ : V × V → R is Hermitian if there exists σ such that :\varphi(x, y) = \sigma(\varphi(y, x)) for all x, y in V. A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism σ is an involution (i.e. of order 2).

Since for an antiautomorphism σ we have for all s, t in R, if , then R must be commutative and φ is a bilinear form. In particular, if, in this case, R is a skewfield, then R is a field and V is a vector space with a bilinear form.

An antiautomorphism σ : R → R can also be viewed as an isomorphism R → Rop, where Rop is the opposite ring of R, which has the same underlying set and the same addition, but whose multiplication operation (∗) is defined by , where the product on the right is the product in R. It follows from this that a right (left) R-module V can be turned into a left (right) Rop-module, Vo. Thus, the sesquilinear form φ : V × V → R can be viewed as a bilinear form φ′ : V × Vo → R.

Notes

References

References

1. also see {{harvnb. Gruenberg. Weir. 1977
2. footnote 1 in [https://books.google.com/books?id=NSXCaGSVaX4C&dq=sesquilinear+forms+over+general+fields&pg=PA255 Anthony Knapp ''Basic Algebra'' (2007) pg. 255]
3. When {{math
4. (1936). "The logic of quantum mechanics". Annals of Mathematics.
5. Baer, Reinhold. (2005). "Linear Algebra and Projective Geometry". Dover.
6. Baer's terminology gives a third way to refer to these ideas, so he must be read with caution.
7. Baer, Reinhold. (2005). "Linear Algebra and Projective Geometry". Dover.
8. ''A'' as an abelian group linear over {{math. ''F'', meaning an [[Module (mathematics). {{math. ''F''-module]] (more colloquially known as a vector space over a skewfield).
9. (2000). "Modern Projective Geometry". [[Kluwer Academic Publishers]].
10. {{harvnb. Jacobson. 2009