division algebra

division algebra

Let

K

be a

unital ring

and

A

a

K

-

algebra

. Defining “division”
requires special considerations when the algebras are

non-associative

so we introduce the definition in stages.

1

Associative division algebras

If

A

is an

associative algebra

then we say

A

is a

division algebra

if

(i)

A

is unital with

identity

1

. So for all

a

∈

A

,

a

⁢

1

=

1

⁢

a

=

a

.

(ii)

Also every non-zero element of

A

has an

inverse

. That is

a

∈

A

,

a

≠

0

, then there exists a

b

∈

A

such that

a

⁢

b

=

1

=

b

⁢

a

.

We denote

b

by

a

-

1

and we may prove

a

-

1

is unique to

a

.

The standard examples of associative division algebras are fields, which
are

commutative

, and the non-split

quaternion algebra

:

α

,

β

∈

K

,

(

α

,

β

K

)

=

{

a

1

1

+

a

2

i

+

a

3

j

+

a

4

k

:

i

2

=

α

1

,

j

2

=

β

1

,

k

2

=

-

α

β

1

,

i

j

=

k

=

-

j

i

.

}

where

x

2

-

α

and

x

2

-

β

are

irreducible

over

K

.

2

Non-associative division algebras

For

non-associative algebras

A

, the notion of an inverse is not immediate.
We use

x

.

y

for the

product

of

x

,

y

∈

A

.

Invertible

as

endomorphisms

:

Let

a

∈

A

. Then define

L

a

:

x

↦

a

.

x

and

R

a

:

x

↦

x

.

a

. As the product of

A

is

distributive

, both

L

a

an

R

a

are

additive

endomorphisms of

A

. If

L

a

is invertible then we may call

a

“

left invertible

” and similarly, when

R

a

is invertible we may call

a

“right invertible” and “invertible” if both

L

a

and

R

a

are invertible.

In this model of invertible,

A

is a

division algebra

if, and only if,
for each non-zero

a

∈

A

, both

L

a

and

R

a

invertible.
Equivalently: the

equations

a

.

x

=

b

and

y

.

a

=

b

have unique solutions
for nonzero

a

,

b

∈

A

. However,

x

and

y

need not be equal.

A common method to produce non-associative division algebras of this sort is
through Schur’s Lemma.

Invertible in the product:

In some

instances

, the notion of invertible via endomorphisms is not

sufficient

. Instead, assume

A

has an identity, that is, an

element

1

∈

A

such
that for all

a

∈

A

,

1

.

a

=

a

=

a

⁢

.1

.

Next if

a

∈

A

, we say

a

is

invertible

if there exists a

b

∈

A

such that

a

.

b

=

1

=

b

.

a

(1)

and furthermore that for all

x

∈

A

,

b

.

(

a

.

x

)

=

x

=

(

x

.

a

)

.

b

.

(2)

Evidently (

1

) can be inferred from (

2

).
This added

assumption

substitutes for the need of associativity in the
proofs of uniqueness of inverses and in solving equations with non-associative
products.

Proposition 1

.

If

A

is a finite dimensional algebra over a field, then
invertible in this sense forces both

L

a

and

R

a

to be invertible as well.

Proof.

Let

x

∈

A

. Then

x

L

1

=

1

.

x

=

x

=

b

.

(

a

.

x

)

=

x

L

a

L

b

. So

L

1

=

L

a

⁢

L

b

. As

L

1

is the

identity map

,

L

a

is

injective

and

L

b

is

surjective

.
As

A

is finite dimensional, injective and surjective endomorphisms are

bijective

.
∎

In this model, a non-associative algebra is a division algebra

A

if it is
unital and every non-zero element is invertible.

3

Alternative division algebras

The standard examples of non-associative division algebras are actually
alternative alegbras, specfically, the

composition algebras

of fields,
non-split quaternions and non-split

octonions

– only the latter are
actually not associative. Invertible in the octonions is interpreted
in the second stronger form.

Theorem 2

(Bruck-Klienfeld).

Every alternative division algebra is either associative or a non-split
octonion.

This result is usually followed by two useful results which serve to omit
the need to consider non-associative examples.

Theorem 3

(Artin-Zorn, Wedderburn).

A finite alternative division algebra is associative and commutative, so
it is a finite field.

Theorem 4

.

An alternative division algebra over an algebraically closed field is
the field itself.

Title

division algebra

Canonical name

DivisionAlgebra

Date of creation

2013-03-22 16:52:03

Last modified on

2013-03-22 16:52:03

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 16K99

Related topic

octonion

Related topic

Octonion

Defines

division algebra

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