Sesquilinear form

Scalar-valued function of two complex variables that is linear in one variable and conjugate-linear in the other

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.

A motivating special case is a sesquilinear form on a complex vector space, V. This is a map V × VC that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism.

An application in projective geometry requires that the scalars come from a division ring (skew field), K, and this means that the "vectors" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary rings R.

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

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[1] and originates in Dirac's bra–ket notation in quantum mechanics.

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.

A complex sesquilinear form can also be viewed as a complex bilinear map

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

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

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.

A σ-sesquilinear form over a right K-module M is a bi-additive map φ : M × MK 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,

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

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

Similarly, xM is orthogonal to yM with respect to φ, written xφ y (or simply xy if φ can be inferred from the context), when φ(x, y) = 0. This relation need not be symmetric, i.e. xy does not imply yx (but see § Reflexivity below).

That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric.

A σ-sesquilinear form φ is called (σ, ε)-Hermitian if there exists ε in K such that, for all x, y in M,

If ε = 1, the form is called σ-Hermitian, and if ε = −1, it is called σ-anti-Hermitian. (When σ is implied, respectively simply Hermitian or anti-Hermitian.)

It also follows that φ(x, x) is a fixed point of the map ασ(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.[2][3][4][5]

In the special case that σ is the identity map (i.e., σ = id), K is commutative, φ is a bilinear form and ε2 = 1. Then for ε = 1 the bilinear form is called symmetric, and for ε = -1 is called skew-symmetric.[6]

Let V be the three dimensional vector space over the finite field F = GF(q2), where q is a prime power. With respect to the standard basis we can write x = (x1, x2, x3) and y = (y1, y2, y3) and define the map φ by:

The map σ : ttq is an involutory automorphism of F. The map φ is then a σ-sesquilinear form. The matrix Mφ associated to this form is the identity matrix. This is a Hermitian form.

In a projective geometry G, a permutation δ of the subspaces that inverts inclusion, i.e.

is called a correlation. A result of Birkhoff and von Neumann (1936)[7] shows that the correlations of desarguesian projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space.[5] A sesquilinear form φ is nondegenerate if φ(x, y) = 0 for all y in V (if and) only if x = 0.

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.[8] (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.)[9]

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.

An element x is orthogonal to another element y with respect to the sesquilinear form φ (written xy) if φ(x, y) = 0. This relation need not be symmetric, i.e. xy does not imply yx.

A sesquilinear form φ : V × VR is reflexive (or orthosymmetric) if φ(x, y) = 0 implies φ(y, x) = 0 for all x, y in V.

A sesquilinear form φ : V × VR is Hermitian if there exists σ such that[10]: 325 

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 σ(st) = σ(t)σ(s) for all s, t in R, if σ = id, 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 : RR can also be viewed as an isomorphism RRop, 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 ab = ba, 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.[11] Thus, the sesquilinear form φ : V × VR can be viewed as a bilinear form φ′ : V × VoR.