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.
This case arises naturally in mathematical physics applications.
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.
Hermitian forms are commonly seen in physics, as the inner product on a complex Hilbert space.
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.
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.
This is the convention used mostly by physicists[1] and originates in Dirac's bra–ket notation in quantum mechanics.
By the universal property of tensor products these are in one-to-one correspondence with complex linear maps
is a finite-dimensional complex vector space, then relative to any basis
More generally, the inner product on any complex Hilbert space is a Hermitian form.
A complex Hermitian form applied to a single vector
Every complex skew-Hermitian form can be written as the imaginary unit
A complex skew-Hermitian form applied to a single vector
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 × 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, The associated anti-automorphism σ for any nonzero sesquilinear form φ is uniquely determined by φ.
(When σ is implied, respectively simply Hermitian or anti-Hermitian.)
[6] Let V be the three dimensional vector space over the finite field F = GF(q2), where q is a prime power.
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.
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.
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 a ∗ b = 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.