|
Let V be a vector space of dimension n over a field K.
If σ is an automorphism of K, a σ-sesquilinear form
on the vector space V over K is a map β : V x V to K such that
eqalignno(
β(u1 + u2, v) &= β(u1, v) + β(u2, v),
β(u, v1 + v2) &= β(u, v1) + β(u, v2)
noalign(rlap(and))
β(au, bv) &= aσ(b)β(u, v).)
for all u, u1, u2, v, v1, v2∈V and all a, b∈K. If σ is the
identity, the form is said to be bilinear.
A linear transformation g of V is an isometry if g preserves
β; it is a similarity if it preserves β up to a non-zero
scalar multiple.
A σ-sesquilinear form β is reflexive if for all u, v∈V,
β(u, v) = 0 implies β(v, u) = 0. Any non-zero multiple of a
reflexive form is again reflexive with the same group of isometries. By a
theorem of Brauer [Bra36] (but sometimes referred to as the
Birkhoff--von Neumann theorem), up to a non-zero scalar multiple, there are
three types of non-degenerate reflexive forms:
- Alternating. In this case σ is the identity,
β(u, u) = 0 for all u∈V and consequently β(u, v) =
- β(v, u) for all u, v ∈V. The group of isometries is a
symplectic group.
- Symmetric. In this case σ is the identity and
β(u, v) = β(v, u) for all u, v ∈V. If the characteristic
of K is not two, the group of isometries is an orthogonal
group. If the characteristic is two, the form is either alternating
or pseudo-alternating (see below).
- Hermitian. In this case σ is an automorphism of
order two and β(u, v) = σβ(v, u) for all u, v ∈V.
The group of isometries is a unitary group.
If V is a vector space V and if β is a reflexive form defined on V,
the partially ordered set of totally isotropic subspaces with respect to
β is often referred to as a
polar space. Similarly, there are polar spaces associated with
quadratic forms (see Section Quadratic Forms). But throughout this
chapter by polar space we shall simply mean a vector space furnished with
either a reflexive σ-sesquilinear form or a quadratic form.
See [Bue95, Chap. 2] for an account of polar spaces in a more
general context.
Let K0 be the fixed field of σ. Multiplying an alternating,
symmetric or hermitian form by a non-zero element of K0 leaves the
type of the form unchanged.
However, multiplying an hermitian form by a non-zero element of K produces
a sesquilinear form ξ and an element ε∈K such that
for all u, v ∈V, ξ(v, u) = εσξ(u, v), where
εσ(ε) = 1. In this case ξ is said to be
ε-hermitian.
- Skew-hermitian. A reflexive σ-sesquilinear form is
skew-hermitian if the order of σ is two and ξ(v, u) =
- σξ(v, u) for all u, v∈V. If β is hermitian and if
d∈K is chosen so that d≠σ(d), then e = d - σ(d)
satisfies σ(e) = - e. Thus ξ(u, v) = eβ(u, v) is
skew-hermitian. The group of isometries of ξ coincides with the
group of isometries of β and it is therefore a unitary group.
In the case of fields of characteristic two there is no distinction between
hermitian and skew-hermitian forms and moreover, every alternating form is
symmetric.
- Pseudo-alternating. A symmetric form (in characteristic two)
which is not alternating is said to be pseudo-alternating.
The three types of forms---alternating, symmetric and hermitian---correspond to
the three types of classical groups of isometries: symplectic, orthogonal and
unitary. But this is not quite the whole story because it does not include
orthogonal groups over fields of characteristic two. In order to include these
groups it is necessary to consider quadratic forms in addition to symmetric
bilinear forms.
If β is a bilinear form, a quadratic form with polar form
β is a function Q : V to K such that
eqalignno(
Q(av) &= a2Q(v)
noalign(rlap(and))
β(u, v) &= Q(u + v) - Q(u) - Q(v))
for all u, v∈V and all a∈K. We have β(v, v) = 2Q(v) and
therefore, if the characteristic of K is not two, β determines Q.
We extend the notion of polar space to include vector spaces V with an
associated quadratic form Q. The pair (V, Q) is an orthogonal geometry
and V is a quadratic space.
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|