In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear, quadratic, and Hermitian forms. In the traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues.

In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types, together with their forms over arbitrary fields. this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups.

The term goes back to Hermann Weyl's book The Classical Groups. Among the simple Lie groups, the classical groups are in contrast to the exceptional Lie groups, G2, F4, E6, E7, E8, which share their abstract properties, but not their familiarity.

, and later discusses the more general formulation over arbitrary fields.

Overview

Two closely related usages of the term classical group and their forms over general fields.

For the purposes of this article, the main families are:

• ;

• the orthogonal groups attached to nondegenerate quadratic or symmetric bilinear forms;

• the symplectic groups attached to nondegenerate alternating forms;

• the unitary groups attached to nondegenerate Hermitian forms relative to an involution.

, the connected simple. Their compact real forms.

The standard real, complex, and quaternionic classical groups are given in the following table:

Linear groups

Real and complex special linear groups

, the special linear group is

Its Lie algebra is

matrices.

Forms and automorphism groups

The classical groups are most naturally described as automorphism groups of nondegenerate forms on finite-dimensional vector spaces.

be a finite-dimensional vector space. A bilinear form is a map

that is linear in each variable. A sesquilinear form on a complex vector space is a map

that is conjugate-linear in the first variable and linear in the second.

For quaternionic vector spaces one usually works with right-vector spaces. In that setting the relevant forms are quaternionic Hermitian or quaternionic skew-Hermitian forms, which are conjugate-linear in the first variable and linear in the second.

, its automorphism group is

becomes a matrix group defined by one of the equations

is bilinear or sesquilinear.

is

or, in matrix form,

Symmetric, alternating, Hermitian, and skew-Hermitian forms

is:

• symmetric;

• alternating.

is:

• Hermitian;

• skew-Hermitian.

all such forms are equivalent.

yields a Hermitian form, so the two cases lead to the same isometry groups up to a harmless change of convention. On a quaternionic vector space, by contrast, there are no nonzero bilinear forms, so only the Hermitian and skew-Hermitian cases occur.

Groups preserving bilinear forms

Orthogonal groups

be a nondegenerate symmetric bilinear form.

, one may choose a basis in which

. Its automorphism group is the indefinite orthogonal group

is the special orthogonal group

.

, every nondegenerate symmetric bilinear form is equivalent to the standard form

Its automorphism group is the complex orthogonal group

.

The corresponding Lie algebras are

If

is equivalent to

Writing

one obtains the block form

matrices.

Symplectic groups

, one may choose a basis in which

Its automorphism group is the symplectic group

for these groups.

Its Lie algebra is

Equivalently, every element has block form

symmetric.

Groups preserving sesquilinear forms

Unitary groups

. One may choose a basis in which

.

Its automorphism group is the unitary group

is the special unitary group

.

Its Lie algebra is

If

then the defining relation is equivalent to

In block form,

The special unitary Lie algebra is the traceless subalgebra

Quaternionic groups

For the quaternionic classical groups, it is convenient to identify

by the complex matrix

This extends to an embedding

GL(n,H) and SL(n,H)

consisting of matrices of the form

Its Lie algebra is therefore

of elements of reduced norm 1.

Its Lie algebra is

Sp(p,q)

and let

be a nondegenerate quaternionic Hermitian form. Its isometry group is the quaternionic unitary group

.

and a nondegenerate complex alternating form. Its Lie algebra is

If

with quaternionic blocks, then the defining relation is equivalent to

In block form,

SO*(2n)

and consider the quaternionic skew-Hermitian form

Its isometry group is the real Lie group

.

Equivalently, if

may be realized as the subgroup

is the involution defining this real form.

.

preserving both the symmetric bilinear form with Gram matrix

and the Hermitian form with Gram matrix

Equivalently,

Its Lie algebra is

Classical groups over arbitrary fields

, the classical groups are the groups of linear automorphisms of a finite-dimensional vector space that preserve either no additional structure, or a nondegenerate alternating, quadratic, or hermitian form. these recover the familiar classical Lie groups, while over a finite field their groups of rational points give the finite classical groups.

Linear groups

. The general linear group is

and the special linear group is

. Their projective quotients are the projective general linear group and the projective special linear group.

Groups preserving forms

The other classical groups arise as automorphism groups of nondegenerate forms.

, its isometry group is the symplectic group

after a choice of basis.

, its isometry group is the orthogonal group

, this is equivalently the group preserving the associated symmetric bilinear form. In characteristic2, orthogonal groups are still defined from quadratic forms, but the relation with the associated bilinear form is subtler.. In the isotropic case and in characteristic not 2, it may be described as the kernel of the spinor norm.

is a quadratic field extension, its isometry group is a unitary group

Its derived subgroup is the special unitary group.

One also has the corresponding similitude, whose elements preserve the relevant form up to multiplication by a scalar. Projective versions are obtained by quotienting by the center.

Classical groups as algebraic groups

In the language of algebraic geometry, a linear algebraic group., together with their forms over fields that are not algebraically closed.

The split classical groups are represented by the following standard examples:

• ;

• ;

• ;

• .

Over a general field, one obtains additional classical groups as inner or outer forms, and many orthogonal or symplectic groups are classified by quadratic or hermitian forms.

, and the finite orthogonal groups.

Classical groups from central simple algebras with involution

is not a split simple algebra. To treat the remaining classical groups, one replaces vector spaces over a field by modules over a central simple algebra with involution.

Over a finite field, this central simple algebra machinery does not produce additional classical groups beyond the usual matrix groups, because every central simple algebra over a finite field is split. Thus the finite classical groups may be described in the language of algebras with involution, but no genuinely non-split examples arise in that setting.

The complete theory of algebras with involution also uses quadratic pairs in the orthogonal case; that extra formalism is only needed to treat characteristic 2. is a field of characteristic different from two.

Involutions and the three types

be an involution. There are two basic cases.

is said to be of the first kind, respectively:

• orthogonal involutions;

• symplectic involutions.

is a quadratic étale is said to be unitary or of the second kind.

This trichotomy corresponds to the three classical families beyond the general linear group:

• ;

• ;

• .

The groups attached to (A,τ)

, we write

for the group of invertible elements, and

for the kernel of the reduced norm.

is of the first kind, then

is the group of isometries, and

is the group of similitudes is called the multiplier of the similitude. (More generally, one can first define these by their associated group schemes.)

, we write:

• in the orthogonal case;

• in the symplectic case.

, then

1. ,

2. ,

and

is denoted

and gives the semisimple simply connected group of unitary type.

the corresponding adjoint form.

Recovery of the split constructions

The field-valued groups discussed earlier are recovered when the algebra is split.

, then

1. ,

and one recovers the ordinary symplectic group.

, then

1. ,

and one recovers the ordinary orthogonal group.

, then

and

1. ,

so one recovers the usual unitary and special unitary groups.

. If the quadratic étale algebra is split,

with the exchange involution. In that case

.

Orthogonal type and Clifford algebras

On the orthogonal side, the structure of the group is governed by the associated Clifford algebra-component, and the corresponding simply connected cover is the spin group. In the split case this recovers the ordinary groups

and

1. .

In the classification of real (and local, this governs the usual passage from a quadratic form to its even Clifford algebra and spin group.

Real forms recovered from the central simple algebra viewpoint

.

In the following table, the labels split and quaternionic refer to the underlying central simple algebra, not necessarily to the resulting real algebraic group. Thus split, while quaternionic. The labels inner and outer: inner, whereas outer.-vector spaces alone.

Examples over local fields

, and for several other standard local fields, the only central division algebras admitting involution of the first kind are the field itself and quaternion algebras. Thus over a local field the first classical groups not obtained from ordinary field-valued forms already require the central simple algebra viewpoint, but at least broadly the classification is similar to that over the real field.

Typical examples are:

• ;

• ;

• .

Notes

References

• E. Artin (1957) via Internet Archive