Classical group

Short description: Type of group in mathematics

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.[1][2]

In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types An, Bn, Cn, and Dn, together with their forms over arbitrary fields.[3][4] Over ℝ and ℂ this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups.[5][6]

The term goes back to Hermann Weyl's book The Classical Groups.[7] 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.[8]

This article begins with the classical Lie groups over ℝ, ℂ, and ℍ, and later discusses the more general formulation over arbitrary fields.

## Contents

* 1 Overview
* 2 Linear groups
* 2.1 Real and complex special linear groups
* 3 Forms and automorphism groups
* 3.1 Symmetric, alternating, Hermitian, and skew-Hermitian forms
* 4 Groups preserving bilinear forms
* 4.1 Orthogonal groups
* 4.2 Symplectic groups
* 5 Groups preserving sesquilinear forms
* 5.1 Unitary groups
* 6 Quaternionic groups
* 6.1 GL(n,H) and SL(n,H)
* 6.2 Sp(p,q)
* 6.3 SO*(2n)
* 7 Classical groups over arbitrary fields
* 7.1 Linear groups
* 7.2 Groups preserving forms
* 7.3 Classical groups as algebraic groups
* 8 Classical groups from central simple algebras with involution
* 8.1 Involutions and the three types
* 8.2 The groups attached to (A,τ)
* 8.3 Recovery of the split constructions
* 8.4 Orthogonal type and Clifford algebras
* 8.5 Real forms recovered from the central simple algebra viewpoint
* 8.6 Examples over local fields
* 9 Notes
* 10 References

## 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 ℝ, ℂ, and ℍ 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 A, B, C, and D and their forms over general fields.[9][10]

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

* the linear groups GL(V) and SL(V);
* 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.[11][12]

Over ℂ, the connected simple classical Lie groups are the families of types An, Bn, Cn, and Dn. Their compact real forms are SU(n), SO(n), and Sp(n).[13]

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  | SL(n,ℝ) | ℝ | —  | SO(n) | 𝔰𝔩n(ℝ) | An−1
Complex special linear  | SL(n,ℂ) | ℂ | —  | SU(n) | 𝔰𝔩n(ℂ) | An−1
Quaternionic special linear  | SL(n,ℍ)≅SU*(2n) | ℍ | —  | Sp(n) | 𝔰𝔲*(2n) | A2n−1
(Indefinite) special orthogonal  | SO(p,q) | ℝ | Symmetric bilinear  | S(O(p)×O(q)) | 𝔰𝔬(p,q) | Bm if p+q=2m+1;
Dm if p+q=2m
Complex special orthogonal  | SO(n,ℂ) | ℂ | Symmetric bilinear  | SO(n) | 𝔰𝔬n(ℂ) | Bm if n=2m+1;
Dm if n=2m
Real symplectic  | Sp(2n,ℝ) | ℝ | Alternating bilinear  | U(n) | 𝔰𝔭2n(ℝ) | Cn
Complex symplectic  | Sp(2n,ℂ) | ℂ | Alternating bilinear  | Sp(n) | 𝔰𝔭2n(ℂ) | Cn
(Indefinite) special unitary  | SU(p,q) | ℂ | Hermitian  | S(U(p)×U(q)) | 𝔰𝔲(p,q) | Ap+q−1
(Indefinite) quaternionic unitary  | Sp(p,q) | ℍ | Hermitian  | Sp(p)×Sp(q) | 𝔰𝔭(p,q) | Cp+q
Quaternionic orthogonal  | SO*(2n) | ℍ | Skew-Hermitian  | U(n) | 𝔰𝔬*(2n) | Dn

## Linear groups

### Real and complex special linear groups

For F=ℝ or ℂ, the special linear group is

SL(n,F)={g∈GLn(F)∣det⁡g=1}.

Its Lie algebra is

𝔰𝔩(n,F)={X∈Mn(F)∣tr⁡(X)=0}.

Thus 𝔰𝔩(n,ℝ) consists of all real traceless n×n matrices, and 𝔰𝔩(n,ℂ) consists of all complex traceless n×n matrices.

## Forms and automorphism groups

Main articles: Bilinear form and Sesquilinear form

The classical groups are most naturally described as automorphism groups of nondegenerate forms on finite-dimensional vector spaces.[14][15]

Let V be a finite-dimensional vector space over ℝ or ℂ. A bilinear form on V is a map

φ:V×V->F

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

φ:V×V->ℂ

that is conjugate-linear in the first variable and linear in the second.[16]

For quaternionic vector spaces one usually works with right ℍ-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.[17]

If φ is a nondegenerate form on V, its automorphism group is

Aut⁡(φ)={g∈GL⁡(V)∣φ(gv,gw)=φ(v,w) for all v,w∈V}.

After a choice of basis, φ is represented by a Gram matrix Φ, and Aut⁡(φ) becomes a matrix group defined by one of the equations

gTΦg=Φ,g*Φg=Φ,

according to whether φ is bilinear or sesquilinear.[18]

The Lie algebra of Aut⁡(φ) is

𝔞𝔲𝔱(φ)={X∈End⁡(V)∣φ(Xv,w)+φ(v,Xw)=0 for all v,w∈V},

or, in matrix form,

XTΦ+ΦX=0,X*Φ+ΦX=0.[19]

### Symmetric, alternating, Hermitian, and skew-Hermitian forms

A bilinear form φ is:

* symmetric if φ(v,w)=φ(w,v);
* alternating (or skew-symmetric, when char⁡F≠2) if φ(v,v)=0 for all v, equivalently φ(v,w)=−φ(w,v).[20]

A sesquilinear form h is:

* Hermitian if h(v,w)=h(w,v)‾;
* skew-Hermitian if h(v,w)=−h(w,v)‾.[21]

Over ℝ, nondegenerate symmetric bilinear forms are classified by their signature (p,q). Over ℂ, all nondegenerate symmetric bilinear forms of a given dimension are equivalent. Nondegenerate alternating forms exist only in even dimension, and over both ℝ and ℂ all such forms are equivalent.[22]

On a complex vector space, multiplying a skew-Hermitian form by i yields a Hermitian form, so the two cases lead to the same isometry groups up to a harmless change of convention.[23] On a quaternionic vector space, by contrast, there are no nonzero bilinear forms, so only the Hermitian and skew-Hermitian cases occur.[24]

## Groups preserving bilinear forms

### Orthogonal groups

Main articles: Orthogonal group, Indefinite orthogonal group, and Complex orthogonal group

Let φ be a nondegenerate symmetric bilinear form.

Over ℝ, one may choose a basis in which

φ(x,y)=xTIp,qy,Ip,q=diag⁡(Ip,−Iq),

where p+q=n. Its automorphism group is the indefinite orthogonal group

O(p,q)={g∈GLn(ℝ)∣gTIp,qg=Ip,q}.

The subgroup of determinant 1 is the special orthogonal group

SO(p,q)=O(p,q)∩SLn(ℝ).[25]

When q=0 this is the compact orthogonal group O(n), with determinant-1 subgroup SO(n).[26]

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

φ(x,y)=xTy.

Its automorphism group is the complex orthogonal group

O(n,ℂ)={g∈GLn(ℂ)∣gTg=In},

with determinant-1 subgroup SO(n,ℂ).[27]

The corresponding Lie algebras are

𝔬(p,q)={X∈Mn(ℝ)∣XTIp,q+Ip,qX=0},
𝔬(n,ℂ)=𝔰𝔬(n,ℂ)={X∈Mn(ℂ)∣XT+X=0}.[28]

g=(ABCD)

with block sizes p×p, p×q, q×p, and q×q, then the defining relation for O(p,q) is equivalent to

ATA−CTC=Ip,DTD−BTB=Iq,ATB=CTD.

Writing

X=(PQRS),

one obtains the block form

𝔬(p,q)={(PQQTS)|PT=−P, ST=−S}.

Thus 𝔰𝔬(n,ℂ) is the space of complex skew-symmetric n×n matrices.

### Symplectic groups

Main article: Symplectic group

Let ω be a nondegenerate alternating bilinear form on a vector space of dimension 2n. Over either ℝ or ℂ, one may choose a basis in which

ω(x,y)=xTJny,Jn=(0In−In0).

Its automorphism group is the symplectic group

Sp(2n,F)={g∈GL2n(F)∣gTJng=Jn},F=ℝ,ℂ.

Many authors write Sp(n,ℝ) and Sp(n,ℂ) for these groups.[29]

Its Lie algebra is

𝔰𝔭(2n,F)={X∈M2n(F)∣XTJn+JnX=0}.

Equivalently, every element has block form

(ABC−AT)

with B and C symmetric.[30]

## Groups preserving sesquilinear forms

### Unitary groups

Main articles: Unitary group and Special unitary group

Let h be a nondegenerate Hermitian form on a complex vector space V of dimension n. One may choose a basis in which

h(z,w)=z*Ip,qw,Ip,q=diag⁡(Ip,−Iq),

where p+q=n.[31]

Its automorphism group is the unitary group

U(p,q)={g∈GLn(ℂ)∣g*Ip,qg=Ip,q}.

The subgroup of determinant 1 is the special unitary group

SU(p,q)=U(p,q)∩SLn(ℂ).[32]

When q=0 this is the compact unitary group U(n), with determinant-1 subgroup SU(n).[33]

Its Lie algebra is

𝔲(p,q)={X∈Mn(ℂ)∣X*Ip,q+Ip,qX=0}.[34]

g=(ABCD),

then the defining relation is equivalent to

A*A−C*C=Ip,D*D−B*B=Iq,A*B=C*D.

In block form,

𝔲(p,q)={(PQQ*R)|P*=−P, R*=−R}.

The special unitary Lie algebra is the traceless subalgebra

𝔰𝔲(p,q)={X∈𝔲(p,q)∣tr⁡(X)=0}.

## Quaternionic groups

For the quaternionic classical groups, it is convenient to identify

ℍ=ℂ⊕jℂ

and to represent a quaternion α+jβ by the complex matrix

(α−β‾βα‾).

This extends to an embedding

Mn(ℍ)↪M2n(ℂ),X+jY↦(X−Y‾YX‾).[35]

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

The group GL(n,ℍ) consists of the invertible quaternionic-linear endomorphisms of the right vector space ℍn. Via the complex embedding above it is realized as a real Lie subgroup of GL(2n,ℂ) consisting of matrices of the form

(A−B‾BA‾),A,B∈Mn(ℂ).[36]

Its Lie algebra is therefore

𝔤𝔩(n,ℍ)={(A−B‾BA‾)|A,B∈Mn(ℂ)}.

The corresponding real form of SL(2n,ℂ) is denoted SU*(2n), and as a Lie group it is isomorphic to the group traditionally written SL(n,ℍ), and is the subgroup of GL(n,ℍ) of elements of reduced norm 1.[37]

Its Lie algebra is

𝔰𝔩(n,ℍ)≅𝔰𝔲*(2n)={(A−B‾BA‾)|Re⁡tr⁡(A)=0}.

### Sp(p,q)

Let V=ℍp+q and let

B(w,z)=w*Ip,qz,Ip,q=diag⁡(Ip,−Iq),

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

Sp(p,q)={g∈GL(p+q,ℍ)∣g*Ip,qg=Ip,q}.[38]

When q=0 this is the compact group usually written Sp(n).[39]

Viewed as a subgroup of GL(2n,ℂ), the group Sp(p,q) preserves both a complex Hermitian form of signature (2p,2q) and a nondegenerate complex alternating form.[40] Its Lie algebra is

𝔰𝔭(p,q)={X∈Mp+q(ℍ)∣X*Ip,q+Ip,qX=0}.[41]

g=(ABCD)

with quaternionic blocks, then the defining relation is equivalent to

A*A−C*C=Ip,D*D−B*B=Iq,A*B=C*D.

In block form,

𝔰𝔭(p,q)={(PQQ*R)|P*=−P, R*=−R}.

### SO*(2n)

Let V=ℍn and consider the quaternionic skew-Hermitian form

C(x,y)=x*jy.

Its isometry group is the real Lie group

SO*(2n),

which is a real form of SO(2n,ℂ).[42]

Equivalently, if

Jn=(0In−In0),

then SO*(2n) may be realized as the subgroup

SO*(2n)={g∈SO(2n,ℂ)∣θ(g)=g},

where θ(g)=−Jng‾Jn is the involution defining this real form.[43]

Its Lie algebra is denoted 𝔰𝔬*(2n).[44]

A standard complex realization of SO*(2n) is as the subgroup of GL(2n,ℂ) preserving both the symmetric bilinear form with Gram matrix

S=(0InIn0)

and the Hermitian form with Gram matrix

H=(In00−In).

Equivalently,

SO*(2n)={g∈GL(2n,ℂ)∣gTSg=S, g*Hg=H}.

Its Lie algebra is

𝔰𝔬*(2n)={(AB−B‾A‾)|A*=−A, BT=−B}.

## Classical groups over arbitrary fields

Over a field k, 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.[5][10] Over ℝ and ℂ these recover the familiar classical Lie groups, while over a finite field their groups of rational points give the finite classical groups.[6]

### Linear groups

Let V be an n-dimensional vector space over k. The general linear group of V is

GL(V)=Autk(V),

and the special linear group is

SL(V)=ker⁡(det⁡:GL(V)->k×).

After a choice of basis these become the matrix groups GLn(k) and SLn(k). Their projective quotients are the projective general linear group PGL(V) and the projective special linear group PSL(V).[3]

### Groups preserving forms

The other classical groups arise as automorphism groups of nondegenerate forms.[5][12]

If ω is a nondegenerate alternating bilinear form on V, its isometry group is the symplectic group

Sp(V,ω)={g∈GL(V)∣ω(gv,gw)=ω(v,w) ∀v,w∈V}.

For dim⁡V=2n, this is written Sp2n(k) after a choice of basis.

If q is a nondegenerate quadratic form on V, its isometry group is the orthogonal group

O(V,q)={g∈GL(V)∣q(gv)=q(v) ∀v∈V}.

When char⁡(k)≠2, 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.[12] For orthogonal groups over general fields, one often also considers the subgroup Ω(V,q). In the isotropic case and in characteristic not 2, it may be described as the kernel of the spinor norm, a homomorphism from SO(V,q) (or more generally from the appropriate index-2 subgroup of O(V,q)) to k×/(k×)2. In the theory of finite classical groups, the simple group is often PΩ(V,q) rather than PSO(V,q).[45]

If K/k is a quadratic field extension, or more generally if K is equipped with an involution σ, and h is a nondegenerate σ-hermitian form on a finite-dimensional K-vector space V, its isometry group is a unitary group

U(V,h)={g∈GLK(V)∣h(gv,gw)=h(v,w) ∀v,w∈V}.

Its derived subgroup is the special unitary group SU(V,h).[12]

One also has the corresponding similitude groups GSp, GO, and GU, whose elements preserve the relevant form up to multiplication by a scalar. Projective versions are obtained by quotienting by the center.[4]

### Classical groups as algebraic groups

In the language of algebraic geometry, a linear algebraic group over k is a smooth affine k-group scheme, equivalently a smooth closed k-subgroup of some GLn.[4][3] From this point of view, the connected classical groups are the connected reductive groups of Dynkin types An, Bn, Cn, and Dn, together with their forms over fields that are not algebraically closed.[4]

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

* type An: SLn+1 and PGLn+1;
* type Bn: SO2n+1 and its simply connected cover Spin2n+1;
* type Cn: Sp2n and PSp2n;
* type Dn: SO2n and Spin2n.[4][3]

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 An, and many orthogonal or symplectic groups are classified by quadratic or hermitian forms.[12][4]

When k=𝔽q is a finite field, the groups of k-rational points of these algebraic groups yield the finite groups of Lie type. The classical families include groups such as PSLn(q), PSUn(q), PSp2n(q), and the finite orthogonal groups.[6]

## 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 ℝ, since ℍ 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 k, or, in the unitary case, over a quadratic field extension of k.[46][47]

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.[46] Henceforth, k is a field of characteristic different from two.

### Involutions and the three types

Let A be a central simple algebra over k, and let τ:A->A be an involution. There are two basic cases.[48]

If τ acts trivially on the center of A, then τ is said to be of the first kind. In characteristic different from two, involutions of the first kind are divided into two types, depending on whether they become adjoints of symmetric or alternating forms after tensoring A with a separable closure of k, respectively:

* orthogonal involutions;
* symplectic involutions.[48]

If the center of A is a quadratic étale k-algebra K and τ induces the nontrivial k-automorphism of K, then τ is said to be unitary or of the second kind.[48]

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

* orthogonal type for Dynkin types Bn and Dn;
* symplectic type for Dynkin type Cn;
* unitary type for outer forms of Dynkin type An.[48][49]

### The groups attached to (A,τ)

For any central simple algebra A, we write

GL1(A)=A×

for the group of invertible elements, and

SL1(A)=ker⁡(Nrd⁡:A×->k×)

for the kernel of the reduced norm. These give the inner forms of type A.[49]

If (A,τ) is of the first kind, then

Iso⁡(A,τ)={a∈A×∣τ(a)a=1}

is the group of isometries, and

Sim⁡(A,τ)={a∈A×∣τ(a)a∈k×}

is the group of similitudes. The scalar μ(a)=τ(a)a is called the multiplier of the similitude.[48] (More generally, one can first define these by their associated group schemes.[48])

According to the type of τ, we write:

* O(A,τ)=Iso⁡(A,τ), GO(A,τ)=Sim⁡(A,τ), and PGO(A,τ)=Aut⁡(A,τ) in the orthogonal case;
* Sp(A,τ)=Iso⁡(A,τ), GSp(A,τ)=Sim⁡(A,τ), and PGSp(A,τ)=Aut⁡(A,τ) in the symplectic case.[48]

If (B,τ) is unitary, with center a quadratic étale algebra K/k, then

U(B,τ)={b∈B×∣τ(b)b=1},
GU(B,τ)={b∈B×∣τ(b)b∈k×},

and

PGU(B,τ)=AutK(B,τ).

The kernel of the reduced norm on U(B,τ) is denoted

SU(B,τ)=ker⁡(Nrd⁡:U(B,τ)->K×),

and gives the semisimple simply connected group of unitary type.[48]

In the symplectic case, Sp(A,τ) is the simply connected group and PGSp(A,τ) its adjoint quotient; in the unitary case, SU(B,τ) is the simply connected form and PGU(B,τ) the corresponding adjoint form.[48][49]

### Recovery of the split constructions

The field-valued groups discussed earlier are recovered when the algebra is split.[48]

If A=Endk(V) and τ is the adjoint involution of a nondegenerate alternating bilinear form h on V, then

Sp(A,τ)=Sp(V,h),

and one recovers the ordinary symplectic group.[48]

If A=Endk(V) and τ is the adjoint involution of a nondegenerate symmetric bilinear form, equivalently of a nondegenerate quadratic form q, then

O(A,τ)=O(V,q),

and one recovers the ordinary orthogonal group.[48]

If K/k is a quadratic field extension, B=EndK(V), and τ is adjoint to a nondegenerate Hermitian form on the K-space V, then

U(B,τ)=U(V,h)

and

SU(B,τ)=SU(V,h),

so one recovers the usual unitary and special unitary groups.[48]

There is also a split description of the inner forms of type A. If the quadratic étale algebra is split,

K≅k×k,

then B is isomorphic to A×Aop with the exchange involution. In that case

U(B,τ)≅GL1(A),SU(B,τ)≅SL1(A),PGU(B,τ)≅PGL1(A).

Thus the same formalism includes both the inner and outer forms of type A.[48]

### Orthogonal type and Clifford algebras

On the orthogonal side, the structure of the group is governed by the associated Clifford algebra. For an orthogonal involution (A,τ), one has a discriminant and a Clifford algebra; in even degree, the center of the even Clifford algebra determines the analogue of the usual +-component, and the corresponding simply connected cover is the spin group. In the split case this recovers the ordinary groups

PGO+(V,q)

and

Spin(V,q).[50]

In the classification of real (and local) classical groups, the orthogonal data require knowing both the algebra A and the involution. (And, if one wants the simply connected groups, the corresponding Clifford algebra.) In characteristic different from 2, this governs the usual passage from a quadratic form to its even Clifford algebra and spin group.[50]

### Real forms recovered from the central simple algebra viewpoint

Over ℝ, the algebra-with-involution framework recovers all of the classical real Lie groups. In particular, the quaternionic families arise only after allowing the noncommutative central simple algebra ℍ.[51][49]

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 means that the algebra is a full matrix algebra over ℝ, while quaternionic means that the algebra is a matrix algebra over ℍ. The labels inner and outer are used only in type A: inner means an inner form of the split group of type A, arising from a central simple ℝ-algebra with center ℝ, whereas outer means a unitary form arising from the quadratic extension ℂ/ℝ.

Dynkin type  | Data over ℝ | Resulting real group

|
|

An−1 (inner, split)  | A=Mn(ℝ) | SLn(ℝ)
A2m−1 (inner, quaternionic)  | A=Mm(ℍ) | SLm(ℍ)≅SU*(2m)
An−1 (outer if n≥3)  | K=ℂ and a Hermitian form of signature (p,q) | SU(p,q) (compact case SU(n)=SU(n,0))
Cn (split)  | a symplectic involution on M2n(ℝ) | Sp2n(ℝ)
Cn (quaternionic)  | a quaternionic Hermitian form of signature (p,q) | Sp(p,q) (compact case Sp(n)=Sp(n,0))
Bn, Dn (split)  | a quadratic form over ℝ of signature (p,q) | SO(p,q) and its spin double cover
Dn (quaternionic)  | a quaternionic skew-Hermitian form on ℍn | SO*(2n) and the corresponding spin group

Combined with the classification of quadratic, Hermitian, and skew-Hermitian forms over ℝ, this gives the standard list of real forms of the classical groups. In The groups SLm(ℍ)≅SU*(2m), Sp(p,q), and SO*(2n) are classical groups over the ground field ℝ even though they are not defined on ordinary ℝ-vector spaces alone.[51][49]

### Examples over local fields

For the real field, for finite extensions of ℚp, 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.[52] 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:

* if F is a finite extension of ℚp and D is the quaternion division algebra over F, then SL1(D) is the inner form of SL2, and more generally SLm(D) is an inner form of SL2m;[49]
* if h is a nondegenerate Hermitian form on a right D-vector space V, and τh is the adjoint involution on EndD(V), then Sp(EndD(V),τh) is a classical group of type Cn; over ℝ with D=ℍ this construction gives the groups Sp(p,q);[49][51]
* if s is a nondegenerate skew-Hermitian form over D, the adjoint involution on EndD(V) is of orthogonal type, and the associated orthogonal and spin groups are nonsplit forms of types Bn or Dn; over ℝ with D=ℍ, the even-dimensional case yields SO*(2n).[49][51]

## Notes

1. ↑ Rossmann 2002
2. ↑ Goodman & Wallach 2009
3. ↑ 3.0 3.1 3.2 3.3 Humphreys, James E. (1975). Linear Algebraic Groups. Springer-Verlag.
4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 Springer, Tonny A. (1998). Linear Algebraic Groups (2nd ed.). Birkhäuser.
5. ↑ 5.0 5.1 5.2 Taylor, Donald E. (1992). The Geometry of the Classical Groups. Heldermann Verlag.
6. ↑ 6.0 6.1 6.2 Humphreys, James E. (2006). Modular Representations of Finite Groups of Lie Type. Cambridge University Press.
7. ↑ Weyl 1939
8. ↑ Wybourne, B. G. (1974). Classical Groups for Physicists, Wiley-Interscience. ISBN 0471965057.
9. ↑ Goodman & Wallach 2009
10. ↑ 10.0 10.1 Dieudonné, Jean (1971). La géométrie des groupes classiques. Springer-Verlag.
11. ↑ Rossmann 2002
12. ↑ 12.0 12.1 12.2 12.3 12.4 Knus, Max-Albert (1991). Quadratic and Hermitian Forms over Rings. Springer-Verlag.
13. ↑ Goodman & Wallach 2009
14. ↑ Rossmann 2002 pp. 91–107.
15. ↑ Goodman & Wallach 2009 pp. 1–16.
16. ↑ Rossmann 2002 pp. 91–93.
17. ↑ Goodman & Wallach 2009 pp. 9–11.
18. ↑ Rossmann 2002 pp. 92–93.
19. ↑ Rossmann 2002 pp. 91–93.
20. ↑ Rossmann 2002 pp. 104–107.
21. ↑ Goodman & Wallach 2009 pp. 9–11.
22. ↑ Rossmann 2002 pp. 104–107.
23. ↑ Goodman & Wallach 2009 p. 84.
24. ↑ Goodman & Wallach 2009 p. 84.
25. ↑ Rossmann 2002 pp. 106–109.
26. ↑ Rossmann 2002 pp. 106–109.
27. ↑ Rossmann 2002 pp. 110–111.
28. ↑ Rossmann 2002 pp. 108–111.
29. ↑ Rossmann 2002 pp. 109–110.
30. ↑ Rossmann 2002 pp. 109–110.
31. ↑ Rossmann 2002 pp. 111–113.
32. ↑ Rossmann 2002 pp. 111–113.
33. ↑ Rossmann 2002 pp. 111–113.
34. ↑ Rossmann 2002 pp. 111–113.
35. ↑ Rossmann 2002 pp. 94–95.
36. ↑ Rossmann 2002 pp. 94–95.
37. ↑ Goodman & Wallach 2009 pp. 7–9, 84–86.
38. ↑ Goodman & Wallach 2009 pp. 9–10.
39. ↑ Goodman & Wallach 2009 p. 84.
40. ↑ Goodman & Wallach 2009 pp. 10–11.
41. ↑ Goodman & Wallach 2009 pp. 15–16.
42. ↑ Goodman & Wallach 2009 pp. 10–11.
43. ↑ Goodman & Wallach 2009 pp. 10–11.
44. ↑ Goodman & Wallach 2009 pp. 16–17.
45. ↑ "Orthogonal group". Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Orthogonal_group.
46. ↑ 46.0 46.1 Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The Book of Involutions. American Mathematical Society Colloquium Publications. 44. American Mathematical Society. ISBN 978-0-8218-0904-4.
47. ↑ "Linear classical group". Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Linear_classical_group.
48. ↑ 48.00 48.01 48.02 48.03 48.04 48.05 48.06 48.07 48.08 48.09 48.10 48.11 48.12 48.13 Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The Book of Involutions. American Mathematical Society Colloquium Publications. 44. American Mathematical Society. pp. 346–351, 363–368. ISBN 978-0-8218-0904-4.
49. ↑ 49.0 49.1 49.2 49.3 49.4 49.5 49.6 49.7 Milne, James S. (2006). "Algebraic Groups and Arithmetic Groups". §27, pp. 192–200. https://webusers.imj-prg.fr/~christophe.cornut/M2/Ref/Milne_AlgGroups.pdf.
50. ↑ 50.0 50.1 Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998). The Book of Involutions. American Mathematical Society Colloquium Publications. 44. American Mathematical Society. pp. 187, 203–204. ISBN 978-0-8218-0904-4.
51. ↑ 51.0 51.1 51.2 51.3 Goodman & Wallach 2009
52. ↑ Milne, James S. (2006). "Algebraic Groups and Arithmetic Groups". Proposition 27.14 and Theorem 27.16, pp. 197–199. https://webusers.imj-prg.fr/~christophe.cornut/M2/Ref/Milne_AlgGroups.pdf.

## References

* E. Artin (1957) Geometric Algebra, chapters III, IV, & V via Internet Archive
* Dieudonné, Jean (1955), La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 5, Berlin, New York: Springer-Verlag, ISBN 978-0-387-05391-2, https://books.google.com/books?id=AfYZAQAAIAAJ
* Goodman, Roe; Wallach, Nolan R. (2009), Symmetry, Representations, and Invariants, Graduate texts in mathematics, 255, Springer-Verlag, ISBN 978-0-387-79851-6
* Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5.
* Hazewinkel, Michiel, ed. (2001), "Classical group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=C/c022410
* Rossmann, Wulf (2002), Lie Groups - An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
* Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, https://books.google.com/books?isbn=0691057567

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