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 , , , and , together with their forms over arbitrary fields. Over and 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.Wybourne, B. G. (1974). Classical Groups for Physicists, Wiley-Interscience. .
This article begins with the classical Lie groups over , , and , and later discusses the more general formulation over arbitrary fields.
Overview
Two closely related usages of the term classical group occur in the literature. In the older matrix-group literature, classical groups are the linear groups over , , and together with the groups preserving nondegenerate forms on those spaces. In the modern theory of algebraic groups, the phrase usually refers to the groups of types , , , and and their forms over general fields.
For the purposes of this article, the main families are: the linear groups and ; 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.
Over , the connected simple classical Lie groups are the families of types , , , and . Their compact real forms are , , and .
The standard real, complex, and quaternionic classical groups are given in the following table:
Name Group Field Form preserved Maximal compact subgroup Lie algebra Root system Special linear
—
Complex special linear
—
Quaternionic special linear
—
(Indefinite) special orthogonal
Symmetric bilinear
if ; if Complex special orthogonal
Symmetric bilinear
if ; if Real symplectic
Alternating bilinear
Complex symplectic
Alternating bilinear
(Indefinite) special unitary
Hermitian
(Indefinite) quaternionic unitary
Hermitian
Quaternionic orthogonal
Skew-Hermitian
Linear groups
Real and complex special linear groups
For or , the special linear group is
Its Lie algebra is
Thus consists of all real traceless matrices, and consists of all complex traceless matrices.
Forms and automorphism groups
The classical groups are most naturally described as automorphism groups of nondegenerate forms on finite-dimensional vector spaces. pp. 91–107. pp. 1–16.
Let be a finite-dimensional vector space over or . A bilinear form on 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. pp. 91–93.
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. pp. 9–11.
If is a nondegenerate form on , its automorphism group is
After a choice of basis, is represented by a Gram matrix , and becomes a matrix group defined by one of the equations
according to whether is bilinear or sesquilinear. pp. 92–93.
The Lie algebra of is
or, in matrix form, pp. 91–93.
Symmetric, alternating, Hermitian, and skew-Hermitian forms
A bilinear form is: symmetric if ; alternating (or skew-symmetric, when ) if for all , equivalently . pp. 104–107.
A sesquilinear form is: Hermitian if ; skew-Hermitian if . pp. 9–11.
Over , nondegenerate symmetric bilinear forms are classified by their signature . Over , all nondegenerate symmetric bilinear forms of a given dimension are equivalent. Nondegenerate alternating forms exist only in even dimension, and over both and all such forms are equivalent. pp. 104–107.
On a complex vector space, multiplying a skew-Hermitian form by yields a Hermitian form, so the two cases lead to the same isometry groups up to a harmless change of convention. p. 84. On a quaternionic vector space, by contrast, there are no nonzero bilinear forms, so only the Hermitian and skew-Hermitian cases occur. p. 84.
Groups preserving bilinear forms
Orthogonal groups
Let be a nondegenerate symmetric bilinear form.
Over , one may choose a basis in which
where . Its automorphism group is the indefinite orthogonal group
The subgroup of determinant is the special orthogonal group pp. 106–109.
When this is the compact orthogonal group , with determinant- subgroup . pp. 106–109.
Over , every nondegenerate symmetric bilinear form is equivalent to the standard form
Its automorphism group is the complex orthogonal group
with determinant- subgroup . pp. 110–111.
The corresponding Lie algebras are
pp. 108–111.
If
with block sizes , , , and , then the defining relation for is equivalent to
Writing
one obtains the block form
Thus is the space of complex skew-symmetric matrices.
Symplectic groups
Let be a nondegenerate alternating bilinear form on a vector space of dimension . Over either or , one may choose a basis in which
Its automorphism group is the symplectic group
Many authors write and for these groups. pp. 109–110.
Its Lie algebra is
Equivalently, every element has block form
with and symmetric. pp. 109–110.
Groups preserving sesquilinear forms
Unitary groups
Let be a nondegenerate Hermitian form on a complex vector space of dimension . One may choose a basis in which
where . pp. 111–113.
Its automorphism group is the unitary group
The subgroup of determinant is the special unitary group pp. 111–113.
When this is the compact unitary group , with determinant- subgroup . pp. 111–113.
Its Lie algebra is pp. 111–113.
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
and to represent a quaternion by the complex matrix
This extends to an embedding pp. 94–95.
GL(n,H) and SL(n,H)
The group consists of the invertible quaternionic-linear endomorphisms of the right vector space . Via the complex embedding above it is realized as a real Lie subgroup of consisting of matrices of the form pp. 94–95.
Its Lie algebra is therefore
The corresponding real form of is denoted , and as a Lie group it is isomorphic to the group traditionally written , and is the subgroup of of elements of reduced norm 1. pp. 7–9, 84–86.
Its Lie algebra is
Sp(p,q)
Let and let
be a nondegenerate quaternionic Hermitian form. Its isometry group is the quaternionic unitary group pp. 9–10.
When this is the compact group usually written . p. 84.
Viewed as a subgroup of , the group preserves both a complex Hermitian form of signature and a nondegenerate complex alternating form. pp. 10–11. Its Lie algebra is pp. 15–16.
If
with quaternionic blocks, then the defining relation is equivalent to
In block form,
SO*(2n)
Let and consider the quaternionic skew-Hermitian form
Its isometry group is the real Lie group
which is a real form of . pp. 10–11.
Equivalently, if
then may be realized as the subgroup
where is the involution defining this real form. pp. 10–11.
Its Lie algebra is denoted . pp. 16–17.
A standard complex realization of is as the subgroup of 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
Over a field , 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. Over and these recover the familiar classical Lie groups, while over a finite field their groups of rational points give the finite classical groups.
Linear groups
Let be an -dimensional vector space over . The general linear group of is
and the special linear group is
After a choice of basis these become the matrix groups and . 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.
If is a nondegenerate alternating bilinear form on , its isometry group is the symplectic group
For , this is written after a choice of basis.
If is a nondegenerate quadratic form on , its isometry group is the orthogonal group
When , this is equivalently the group preserving the associated symmetric bilinear form. In characteristic 2, orthogonal groups are still defined from quadratic forms, but the relation with the associated bilinear form is subtler. For orthogonal groups over general fields, one often also considers the subgroup . In the isotropic case and in characteristic not 2, it may be described as the kernel of the spinor norm, a homomorphism from (or more generally from the appropriate index-2 subgroup of ) to . In the theory of finite classical groups, the simple group is often rather than .
If is a quadratic field extension, or more generally if is equipped with an involution , and is a nondegenerate -hermitian form on a finite-dimensional -vector space , its isometry group is a unitary group
Its derived subgroup is the special unitary group .
One also has the corresponding similitude groups , , and , 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 over is a smooth affine -group scheme, equivalently a smooth closed -subgroup of some . From this point of view, the connected classical groups are the connected reductive groups of Dynkin types , , , and , together with their forms over fields that are not algebraically closed.
The split classical groups are represented by the following standard examples: type : and ; type : and its simply connected cover ; type : and ; type : and .
Over a general field, one obtains additional classical groups as inner or outer forms of these split groups. For example, unitary groups are outer forms of type , and many orthogonal or symplectic groups are classified by quadratic or hermitian forms.
When is a finite field, the groups of -rational points of these algebraic groups yield the finite groups of Lie type. The classical families include groups such as , , , and the finite orthogonal groups.
Classical groups from central simple algebras with involution The previous section described classical groups attached to vector spaces over a field, together with unitary groups attached to quadratic field extensions. That accounts for the split classical groups and the usual unitary groups, but it does not include the quaternionic families over , since 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. The usual constructions of classical groups in the previous section are recovered when the algebra is a matrix algebra over , or, in the unitary case, over a quadratic field extension of .