free associative algebra

free associative algebra

Fix a

commutative

unital ring

K

and a set

X

. Then a

K

-

algebra

F

is
said to be

free on

X

if there exists an

injection

ι

:

X

→

F

such that for all

functions

f

:

X

→

A

where

A

is an

K

-algebra
determine a unique algebra homomorphism

f

^

:

F

→

A

such that

ι

⁢

f

^

=

f

. This is an example of a

universal mapping property

for

free associative algebras

and in

categorical

settings is often explained
with the following

commutative diagram

:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

l

⁢

d

]

ι

⁢

\ar

⁢

[

r

⁢

d

]

f

⁢

&

⁢

F

⁢

\ar

⁢

[

r

⁢

r

]

f

^

⁢

&

⁢

&

⁢

A

.

To prove that free associative algebras exist in the

category

of all

associative algebras

we provide a couple standard constructions. It
is a standard categorical procedure to conclude any two free objects
on the same set are

naturally equivalent

and thus each construction
below is

equivalent

.

1

Tensor algebra

Let

X

be a set and

K

a commutative unital ring. Then take

M

to be
any free

K

-module with basis

X

, and injection

ι

:

X

→

M

.
Then we may form the

tensor algebra

of

M

,

T

⁢

(

M

)

=

⊕

i

∈

ℕ

T

i

⁢

(

M

)

,

T

i

⁢

(

M

)

=

M

⊗

i

=

⊗

j

=

1

i

M

.

[Note,

0

∈

ℕ

and the empty tensor we define as

K

.]
Furthermore, define the injection

ι

′

:

X

→

T

⁢

(

M

)

as the map

ι

:

X

→

M

followed by the

embedding

of

M

into

T

⁢

(

M

)

.

Remark 1

.

To make

M

concrete use the set of all functions

f

:

X

→

K

, or
equivalently, the

direct product

∏

X

K

. Then the tensor algebra
of

M

is the

free algebra

on

X

.

Proposition 2

.

(

T

⁢

(

M

)

,

ι

′

)

is a free associative algebra on

X

.

Proof.

Given any

associative

K

-algebra

A

and function

f

:

X

→

A

, then

A

is a

K

-module and

M

is free on

X

so

f

extends to a unique

K

-linear

homomorphism

f

^

:

M

→

A

.

Next we define

K

-multilinear maps

f

(

i

)

:

M

i

→

A

by

f

(

i

)

⁢

(

m

1

,

…

,

m

i

)

=

f

⁢

(

m

1

)

⁢

⋯

⁢

f

⁢

(

m

i

)

.

Then by the universal mapping property of

tensor products

(used inductively)
we have a unique

K

-

linear map

f

^

(

i

)

:

T

i

⁢

(

M

)

→

A

for which

f

^

(

i

)

⁢

(

m

1

⊗

⋯

⊗

m

i

)

=

f

⁢

(

m

1

)

⁢

⋯

⁢

f

⁢

(

m

i

)

.

Thus we have a unique algebra homomorphism

f

^

(

∞

)

:

T

⁢

(

M

)

→

A

such that

ι

⁢

f

^

=

f

.
∎

This construction provides an obvious

grading

on the free algebra where
the

homogeneous components

are

T

n

⁢

(

M

)

=

M

⊗

n

=

⊗

j

=

1

n

M

.

2

Non-commutative polynomials

An alternative construction is to model the methods of constructing

free
groups

and semi-groups, that is, to use words on the set

X

. We will
denote the result of this construction by

K

⁢

⟨

X

⟩

and we will
find many

parallels

to

polynomial algebras

with indeterminants in

X

.

Let

F

⁢

M

⁢

⟨

X

⟩

be the set of all words on

X

. This makes

F

⁢

M

⁢

⟨

X

⟩

a

free monoid

with

identity

the

empty word

and
associative

product

the

juxtaposition

of words. Then define

K

⁢

⟨

X

⟩

as the

K

-semi-group algebra on

F

⁢

M

⁢

⟨

X

⟩

.
This means

K

⁢

⟨

X

⟩

is the free

K

-modules oN

F

⁢

M

⁢

⟨

X

⟩

and the product is defined as:

(

∑

w

∈

F

⁢

M

⁢

⟨

X

⟩

l

w

⁢

w

)

⁢

(

∑

v

∈

F

⁢

M

⁢

⟨

X

⟩

l

v

⁢

v

)

=

∑

w

,

v

∈

F

⁢

M

⁢

⟨

X

⟩

l

v

⁢

l

w

⁢

w

⁢

v

.

For example,

ℚ

⁢

⟨

x

,

y

⟩

contains elements of the form

x

2

+

4

⁢

y

⁢

x

⁢

y

,

-

7

⁢

x

⁢

y

+

2

⁢

y

⁢

x

,

1

+

x

+

x

⁢

y

+

x

⁢

y

⁢

x

+

x

2

⁢

y

+

x

2

⁢

y

2

.

This model of a free associative algebra encourages a

mapping

to

polynomial
rings

. Indeed,

K

⁢

⟨

X

⟩

→

K

⁢

[

X

]

is uniquely determined by
the free property applied to the natural

inclusion

of

X

into

K

⁢

[

X

]

.
What we realize this mapping in a practical fashion we note that this simply
allows all indeterminants to commute. It follows from this that

K

⁢

[

X

]

is
a free commutative associaitve algebra.

For example, under this map we

translate

the above

elements

into:

x

2

+

4

⁢

x

⁢

y

2

,

-

5

⁢

x

⁢

y

,

1

+

x

+

x

⁢

y

+

2

⁢

x

2

⁢

y

+

x

2

⁢

y

2

.

We also note that the grading detected in the tensor algebra construction
persists in the non-commuting

polynomial

model. In particular, we say an
element in

K

⁢

⟨

X

⟩

is

homogeneous

if it

contained

in

F

⁢

M

⁢

⟨

X

⟩

. Then the degree of a

homogeneous element

is the
length of the word. Then the

K

-

linear span

of elements of degree

i

form the

i

-th graded

component

of

K

⁢

⟨

X

⟩

.

Remark 3

.

We note that the free properties of both of these constructions depend
in turn on the free properties of modules, the

universal property

of
tensors and free semi-groups. An inspection of the common construction
of tensors and

free modules

reveals both of these have universal properties
implied from the universal mapping property of free semi-groups. Thus
we may assert that free of associative algebras are a direct result of
the

existence

of free semi-groups.

For

non-associative algebras

such as Lie and

Jordan algebras

, the
universal properties are more subtle.

Title

free associative algebra

Canonical name

FreeAssociativeAlgebra

Date of creation

2013-03-22 16:51:07

Last modified on

2013-03-22 16:51:07

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

10

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 08B20

Related topic

Algebras

Related topic

TensorAlgebra

Defines

free associative algebra

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