classical groups

classical groups

1

Classical Groups

It is commonplace to express the

classical groups

with explicit matrices; however, the theory and classification of classical groups can benefit from a basis free consideration.

Given a finite dimensional

vector space

V

over any field

k

, the set of all

linear transformations

on

V

is denoted

G

⁢

L

⁢

(

V

)

and called the

general linear group

. We now define several significant related groups of

G

⁢

L

⁢

(

V

)

.

Theorem 1

(Birkhoff-von Neumann).

Given a

reflexive non-degenerate sesquilinear

form

b

:

V

×

V

→

k

, then
up to a

constant

b

is one of the following:

•

Alternating

: that is

b

⁢

(

v

,

v

)

=

0

.

•

Hermitian

: so

b

⁢

(

v

,

w

)

=

b

⁢

(

w

,

v

)

σ

where

σ

is a field
automorphism of

k

of order

2

.

•

Symmetric

: so

b

⁢

(

v

,

w

)

=

b

⁢

(

w

,

v

)

.

(Refer to

[

3

,

Theorem

7.1]

and

[

1

, Chapter V]

.)
We prefer the

definition

b

⁢

(

v

,

v

)

=

0

over

b

⁢

(

v

,

w

)

=

-

b

⁢

(

w

,

v

)

so that we can accommodate the fields of

characteristic

2. In all other characteristics these two

properties

are

equivalent

.

In keeping with tradition for

group theory

, we let

G

⁢

L

⁢

(

V

)

act on the vector space

V

on the

right hand side

. This means given

v

∈

V

and

f

∈

G

⁢

L

⁢

(

V

)

,

v

⁢

f

corresponds the vector in

V

which

f

sends

v

to. If one thinks of

f

as a matrix this requires

v

to be a

row vector

. It is also common to consider

f

as a

function

and use the notation

f

⁢

(

v

)

.

Definition 2

.

Given a reflexive non-degenerate sesquilinear form

b

:

V

×

V

→

k

we define

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

=

{

f

∈

G

⁢

L

⁢

(

V

)

:

b

⁢

(

v

⁢

f

,

w

⁢

f

)

=

b

⁢

(

v

,

w

)

,

v

,

w

∈

V

}

.

This is called the isometry group of

b

in

G

⁢

L

⁢

(

V

)

.

Proposition 3

.

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

is a

subgroup

of

G

⁢

L

⁢

(

V

)

.

Proof.

Given

f

,

g

∈

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

,

v

,

w

∈

V

then

b

⁢

(

v

⁢

(

f

⁢

g

)

,

w

⁢

(

f

⁢

g

)

)

=

b

⁢

(

(

v

⁢

f

)

⁢

g

,

(

w

⁢

f

)

⁢

g

)

=

b

⁢

(

v

⁢

f

,

w

⁢

f

)

=

b

⁢

(

v

,

w

)

.

Hence

f

⁢

g

∈

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

. Clearly

1

∈

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

as well. Finally,

b

⁢

(

v

⁢

f

-

1

,

w

⁢

f

-

1

)

=

b

⁢

(

(

v

⁢

f

-

1

)

⁢

f

,

(

w

⁢

f

-

1

)

⁢

f

)

=

b

⁢

(

v

,

w

)

.

So

f

-

1

∈

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

and

I

⁢

s

⁢

o

⁢

m

⁢

(

b

)

is a subgroup of

G

⁢

L

⁢

(

V

)

.
∎

Now if we return to Theorem

1

we find that there are only three isometry group types, as there are only three types of reflexive non-degenerate sesquilinear forms. These receive the well-known names:

•

Symplectic group

S

⁢

p

⁢

(

V

,

b

)

if

b

is alternating.

•

Unitary group

U

⁢

(

V

,

b

)

if

b

is hermitian.

•

Orthogonal group

O

⁢

(

V

,

b

)

if

b

is symmetric.

A vector space

V

equipped with a

reflexive

non-degenerated

sesquilinear form

b

is also given the designation symplectic,

unitary

, and

orthogonal

based on the classification of the form.

Because symplectic spaces have a standard

hyperbolic basis

it follows every symplectic group over a vector space of the same

dimension

is isometric, meaning

isomorphic

as vector spaces but with an

isomorphism

which respects the forms. Thus we can write

S

⁢

p

⁢

(

V

)

instead of

S

⁢

p

⁢

(

V

,

b

)

. For unitary and orthogonal groups more care is required.

Definition 4

.

A

classical group

is any one of the family of groups derived from these three and the general linear group.

2

Classical Groups as Matrices

When expressing these

groups with

matrices it becomes necessary to establish the

bilinear forms

with matrices. Given any

n

×

n

-matrix

B

over some field

k

, and row vectors

v

,

w

∈

k

n

we have a reflexive bilinear form defined by

b

⁢

(

v

,

w

)

=

v

⁢

B

⁢

w

t

.

Whence

b

is

non-degenerate

if and only if

det

⁡

B

≠

0

.

The most common example is the

identity matrix

B

=

I

n

. For then

b

⁢

(

v

,

w

)

=

v

⁢

w

t

is the usual

dot product

, only perhaps without the

positive definite

axiom which makes sense only for

ordered fields

like the

rationals

ℚ

and reals

ℝ

.

The isometry group of

I

is nothing more than the

invertible matrices

A

where

A

⁢

I

⁢

A

t

=

I

;

A

⁢

A

t

=

I

.

Thus it is common for

O

⁢

(

n

)

to denote the orthogonal group over

ℝ

and be given by

O

⁢

(

n

)

=

{

A

∈

G

⁢

L

⁢

(

n

)

:

A

⁢

A

t

=

I

}

.

For symplectic groups the form is the typical

J

=

[

0

I

-

I

0

]

matrix found in the definition of symplectic matrices.
Hence the

isometry

condition for an alternating form

b

⁢

(

v

⁢

A

,

w

⁢

A

)

=

v

⁢

A

⁢

J

⁢

A

t

⁢

w

t

=

v

⁢

J

⁢

w

t

=

b

⁢

(

v

,

w

)

show that

A

⁢

J

⁢

A

t

=

J

. Thus it is common to define

S

⁢

p

⁢

(

2

⁢

m

)

=

{

A

∈

G

⁢

L

⁢

(

2

⁢

m

)

:

A

⁢

J

⁢

A

t

=

J

}

.

Thus the symplectic matrices form a group of isometries.

3

Special subgroups

The

commutator

of

G

⁢

L

⁢

(

V

)

is the

special linear group

S

⁢

L

⁢

(

V

)

composed of all

invertible linear transformations

of

determinant

1. Given a reflexive non-degenerate sesquilinear form

b

on

V

, we can create the groups

S

⁢

I

⁢

s

⁢

o

⁢

m

⁢

(

V

,

b

)

=

I

⁢

s

⁢

o

⁢

m

⁢

(

V

,

b

)

∩

S

⁢

L

⁢

(

V

)

.

These get the names

S

⁢

p

⁢

(

V

)

,

S

⁢

U

⁢

(

V

,

b

)

,

and

⁢

S

⁢

O

⁢

(

V

,

b

)

.

Notice that

S

⁢

p

⁢

(

V

)

≤

S

⁢

L

⁢

(

V

)

so we do not require a new name.

4

Projective groups

The

projective geometry

of a vector space

V

, denoted

P

⁢

G

⁢

(

V

)

is its

lattice

of

subspaces

. Clearly

invertible linear maps

act on the projective geometry because they send points (1-dimensional subspaces) to points, and lines (2-dimensional subspaces) to lines, and in general

m

-dimensional subspaces to other

m

-dimensional subspaces.

However, the scalar

transformations

, i.e.: those

f

∈

G

⁢

L

⁢

(

V

)

such that

v

⁢

f

=

v

⁢

λ

for some fixed

λ

∈

k

, do not move any of the subspaces of

V

– they fix

P

⁢

G

⁢

(

V

)

. Therefore when we consider the action of

G

⁢

L

⁢

(

V

)

on

P

⁢

G

⁢

(

V

)

we factor out the kernel of the action – that is the scalar transforms (matrices.) We denote this group by

P

⁢

G

⁢

L

⁢

(

V

)

. Because

scalar matrices

commute with all other, and not other matrices do, we notice this is the same as

factoring

by the center

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

.

Immediately this gives rise the projective versions of each of the classical groups: Let

Z

=

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

– the set of scalar transformations (a group isomorphic to

k

×

.)

•

P

⁢

G

⁢

L

⁢

(

V

)

=

G

⁢

L

⁢

(

V

)

/

Z

,

P

⁢

S

⁢

L

⁢

(

V

)

=

S

⁢

L

⁢

(

V

)

/

(

Z

∩

S

⁢

L

⁢

(

V

)

)

•

P

⁢

S

⁢

p

⁢

(

V

)

=

S

⁢

p

⁢

(

V

)

/

(

Z

∩

S

⁢

p

⁢

(

V

)

)

•

P

⁢

U

⁢

(

V

)

=

U

⁢

(

V

,

b

)

/

(

Z

∩

U

⁢

(

V

,

b

)

)

,

P

⁢

S

⁢

U

⁢

(

V

,

b

)

=

S

⁢

U

⁢

(

V

,

b

)

/

(

Z

∩

S

⁢

U

⁢

(

V

,

b

)

)

•

P

⁢

O

⁢

(

V

)

=

O

⁢

(

V

,

b

)

/

(

Z

∩

O

⁢

(

V

,

b

)

)

,

P

⁢

S

⁢

O

⁢

(

V

,

b

)

=

S

⁢

O

⁢

(

V

,

b

)

/

(

Z

∩

S

⁢

O

⁢

(

V

,

b

)

)

Most of the time the projective special isometry groups are

simple groups

.
The exceptions arise for small dimensional vector spaces and/or small fields, or with the orthogonal groups.

References

1

Gruenberg, K. W. and Weir, A.J.

Linear

Geometry

2nd Ed.

(English)
[B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. (1977), pp. x-198.

2

Kantor, W. M.

Lectures notes on Classical Groups

.

3

Taylor, Donald E.

The geometry of the classical groups

Sigma Series in Pure Mathematics. 9.
Heldermann Verlag, Berlin, xii+229, (1992), ISBN 3-88538-009-9.

Title

classical groups

Canonical

name

ClassicalGroups

Date of creation

2013-03-22 15:50:13

Last modified on

2013-03-22 15:50:13

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

23

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 11E57

Synonym

linear algebraic groups

Related topic

SemilinearTransformation

Related topic

PolaritiesAndForms

Related topic

SesquilinearFormsOverGeneralFields

Defines

classical group

Defines

isometry

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