centralizers in algebra

centralizers in algebra

1

Abstract definitions and properties

Definition 1

.

Let

S

be a set with a

binary operation

*

. Let

T

be a subset of

S

. Then define
the

centralizer

in

S

of

T

as the subset

C

S

⁢

(

T

)

=

{

s

∈

S

:

s

*

t

=

t

*

s

,

for all

⁢

t

∈

T

}

.

The

center

of

S

is defined as

C

S

⁢

(

S

)

. This is commonly denoted

Z

⁢

(

S

)

where

Z

is derived from the German word

zentral

. Subsets and elements of the center
are called

central

.

If we regard

*

:

S

×

S

→

S

in the of actions we can perscribe a

left
action

s

*

t

and a right action

t

*

s

. The centralizer is thus the set of

elements

for which the left

regular

action and the right regular action agree when
to

T

.

It is generally possible to have

s

*

t

not lie in

T

for

s

∈

C

S

⁢

(

T

)

and

t

∈

T

, and likewise, it is also possible that if

s

,

s

′

∈

C

S

⁢

(

T

)

that

s

*

s

′

≠

s

′

*

s

.
Therefore it should not be presumed that the centralizer is central.

With further axioms on the of

operation

we can deduce certain natural
for the set

C

S

⁢

(

T

)

.

Proposition 2

.

1.

If

A

⊆

B

, then

C

S

⁢

(

B

)

⊆

C

S

⁢

(

A

)

. In particular,

C

S

⁢

(

∅

)

=

S

.

2.

If

S

has an

identity

then

C

S

⁢

(

T

)

is non-empty. In particular, in this case

Z

⁢

(

S

)

is non-empty.

1

1

An identity of

S

is an element

e

∈

S

such that

e

*

s

=

s

*

e

for all

s

∈

S

.

3.

If

S

is

associative

and

s

,

s

′

∈

C

S

⁢

(

T

)

then

s

*

s

′

∈

C

S

⁢

(

T

)

,
we say then that

C

S

⁢

(

T

)

is closed to the binary operation of

S

.

4.

If

s

∈

C

S

⁢

(

T

)

and

s

has an (strong)

inverse

s

-

1

, then

s

-

1

∈

C

S

⁢

(

T

)

.

2

2

We say an inverse is strong if

s

-

1

*

(

s

*

t

)

=

t

=

(

t

*

s

)

*

s

-

1

for
all

t

∈

S

. If the operation is associative then this is given for free. There
are natural nonassociative operations with this property, such as

alternative algebras

.

5.

If

S

is

commutative

then

C

S

⁢

(

T

)

=

S

.

6.

If

T

is a subset of the center of

S

then

C

S

⁢

(

T

)

=

S

.

Note that it is possible for

C

S

⁢

(

T

)

be a subset closed to the opertaion without the

assumption

of associativity, as for example, when

S

is commutative.

2

Centralizers in groups

In the category of groups the centralizer in a group

G

of a subset

H

can be redefined as:

C

G

⁢

(

H

)

=

{

g

∈

G

:

g

-

1

⁢

h

⁢

g

=

h

,

for all

⁢

h

∈

H

}

.

If one regards

conjugation

as a

group action

h

g

:=

g

-

1

⁢

h

⁢

g

then it follows that the
centralizer is the same as the

pointwise

stabilizer

in

G

of

H

, where the action is
of

G

on itself by conjugation. Because of this overlap, in some

contexts

the
centralizers is applied to the pointwise stabilizer of a set on which a group acts,
though this context no longer refers to the action of conjugation. This
is espeically common when there is a need to distinguish between the pointwise stabilizer
and the setwise stabilizer.

In this

category

, the centralizer is always a

subgroup

of

G

. Furthermore, if

H

is a

normal
subgroup

of

G

, then so too is

C

G

⁢

(

H

)

.

3

Centralizers in rings and algebras

For we treat rings as algebras over

ℤ

and now speak only of

algebras

, which will include nonassociative examples.

In an algebra

A

there is in fact two binary operations on the set

A

in
question. Thus the abstract definition of the centralizer is ambiguous. However, the

additive

operation of rings and algebras is always commutative and so any centralizer with respect to this
operation is the set

A

. Thus it is generally accepted practice to assume that centralizers
in this context always refer to the multiplicative operation. In this way we have the following
properties.

Proposition 3

.

Given an algebra

A

over a commutative

unital ring

R

and a subset

B

of

A

, then

1.

C

A

⁢

(

B

)

is a

submodule

of

A

.

2.

If

A

is associative then

C

A

⁢

(

B

)

is a

subalgebra

.

3.

Z

⁢

(

A

)

≤

C

A

⁢

(

B

)

, in particular, if

A

has a 1 then

1

∈

C

A

⁢

(

B

)

and so

R

embeds in

C

A

⁢

(

B

)

.

Remarks

.

•

A centralizer in an algebra is also called a commutant. This terminology is mostly used in algebras of

operators

in functional analysis.

•

Let

R

be a ring (or an algebra). For every

ordered pair

(

a

,

b

)

of elements of

R

, we can define the

additive commutator

of

(

a

,

b

)

to be the element

a

⁢

b

-

b

⁢

a

, written

[

a

,

b

]

. With this, one may alternatively define the centralizer of a set

S

⊆

R

in a ring

R

as

C

R

⁢

(

S

)

:=

{

r

∈

R

∣

[

r

,

s

]

=

0

⁢

for all

⁢

s

∈

S

}

.

Of course, in this definition, two operations (

multiplication

and subtraction) are needed instead of one. But the nice aspect about this definition is that one can “measure” commutativity of a ring by the additive commutation operation. For example, one can show that, in a

division ring

, if every element additively commutes with every additive commutator, then the ring must be a field.

4

Centralizers in Lie algebras

Suppose

𝔤

is a Lie aglebra over a

commutative ring

of

characteristic

not

2

.
Given a subset

T

of

𝔤

, then

[

s

,

t

]

=

-

[

t

,

s

]

for

s

∈

𝔤

and

t

∈

T

from the axioms
of a

Lie algebra

multiplication. Therefore whenever

[

s

,

t

]

=

[

t

,

s

]

it follows that

-

[

t

,

s

]

=

[

t

,

s

]

so that

[

t

,

s

]

=

0

. This motivates the more common redefinition of
the centralizer in a Lie algebra:

C

𝔤

⁢

(

T

)

=

{

s

∈

𝔤

:

[

s

,

t

]

=

0

,

for all

⁢

t

∈

T

}

.

Despite the incongruety in characteristic 2, this new definition replaces the original
definition of centralizers for Lie algebras. The centralizer of a Lie algebra is a subalgebra.

When the Lie multiplication is regarded as a

commutator

, so

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

, for example
it it

universal enveloping algebra

, then

0

=

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

is the same as

a

⁢

b

=

b

⁢

a

and so
the centralizer of the Lie algebra coincides with the centralizer of the

associative
envelope

.

5

Centralizers in other nonassociative algebras

The centralizer need not be a subalgebra on account of the lack of associativity.
There are

instances

of

non-associative algebras

where the centralizer is however
a subalgebra nontheless, for example, Lie algebras as seen above. In travial fashion, if
an algebra is commutative then

C

A

⁢

(

B

)

=

A

and so the centralizer is a subalgebra
but without any useful properties. There is a suitable additional constraint
to add to centralizers to force them to be subalgebras and carry with them
more useful in the commutative but nonassociative setting.

We write

[

a

,

b

]

for

a

⁢

b

-

b

⁢

a

, called the commutator in

A

of

a

,

b

∈

A

and also write

/

a

,

b

,

c

/

for

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

and call it the

associator

in

A

of

a

,

b

,

c

∈

A

.

3

3

This notation
for associators is non-standard but the standard

[

a

,

b

,

c

]

=

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

is likely confusing given the
usual commutator notation used already.

Then we
can redefine the centralizer in

A

of a subset

T

of

A

as

C

A

(

T

)

=

{

s

∈

A

:

[

s

,

t

]

=

0

,

/

s

′

,

s

,

t

/

=

/

s

′

,

t

,

s

/

=

/

t

,

s

′

,

s

/

=

0

for all

t

∈

T

,

s

′

∈

A

}

.

It follows that

C

A

⁢

(

T

)

is a subalgebra of

A

on account of the added associator condition
which forces the subset to be closed to the

product

.

In alternative algebras, if any one of three associators is 0 then the other three are as well
and so the definition reduces to

/

s

′

,

s

,

t

/

=

0

. occur of other

nonassociative
algebras

.

Title

centralizers in algebra

Canonical name

CentralizersInAlgebra

Date of creation

2013-03-22 17:22:30

Last modified on

2013-03-22 17:22:30

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

14

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 17A01

Synonym

centraliser

Related topic

CentralizerOfASubsetOfAGroup

Related topic

LieAlgebra

Defines

centralizer

Defines

center

Defines

central

Defines

centraliser

Defines

additive commutator

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