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 .
is a free associative algebra on .
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 .
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.
|Title||free associative algebra|
|Date of creation||2013-03-22 16:51:07|
|Last modified on||2013-03-22 16:51:07|
|Last modified by||Algeboy (12884)|
|Defines||free associative algebra|