free associative algebra
Fix a commutative unital ring and a set . Then a -algebra
is
said to be free on if there exists an injection such that for all functions where is an -algebra
determine a unique algebra homomorphism such that
. 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
:
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 be a set and a commutative unital ring. Then take to be any free -module with basis , and injection . Then we may form the tensor algebra of ,
[Note, and the empty tensor we define as .]
Furthermore, define the injection as the map followed by the embedding of into .
Remark 1.
To make concrete use the set of all functions , or
equivalently, the direct product . Then the tensor algebra
of is the free algebra
on .
Proposition 2.
is a free associative algebra on .
Proof.
Given any associative -algebra and function , then
is a -module and is free on so extends to a unique
-linear homomorphism .
Next we define -multilinear maps by
Then by the universal mapping property of tensor products (used inductively)
we have a unique -linear map for which
Thus we have a unique algebra homomorphism such that . ∎
This construction provides an obvious grading on the free algebra where the homogeneous components are
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 . We will
denote the result of this construction by and we will
find many parallels to polynomial algebras with indeterminants in .
Let be the set of all words on . This makes
a free monoid with identity the empty word and
associative product
the juxtaposition of words. Then define
as the -semi-group algebra on .
This means is the free -modules oN
and the product is defined as:
For example, contains elements of the form
This model of a free associative algebra encourages a mapping to polynomial
rings. Indeed, is uniquely determined by
the free property applied to the natural inclusion of into .
What we realize this mapping in a practical fashion we note that this simply
allows all indeterminants to commute. It follows from this that is
a free commutative associaitve algebra.
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 is homogeneous if it contained in
. Then the degree of a homogeneous element
is the
length of the word. Then the -linear span of elements of degree
form the -th graded component
of .
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 |