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[parent] free associative algebra (Definition)

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 $ \iota:X\rightarrow F$ such that for all functions $ f:X\rightarrow A$ where $ A$ is an $ K$-algebra determine a unique algebra homomorphism $ \hat{f}:F\rightarrow A$ such that $ \iota\hat{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:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X\ar[ld]_{\iota}\ar[rd]^{f} & \ F\ar[rr]^{\hat{f}} & & A. } } \end{xy}$

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.

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 $ \iota:X\rightarrow M$. Then we may form the tensor algebra of $ M$,

$\displaystyle T(M)=\bigoplus_{i\in\mathbb{N}} T^i(M),\qquad T^i(M)=M^{\otimes i}= \bigotimes_{j=1}^i M.$
[Note, $ 0\in\mathbb{N}$ and the empty tensor we define as $ K$.] Furthermore, define the injection $ \iota':X\rightarrow T(M)$ as the map $ \iota:X\rightarrow M$ followed by the embedding of $ M$ into $ T(M)$.
Remark 1   To make $ M$ concrete use the set of all functions $ f:X\rightarrow K$, or equivalently, the direct product $ \prod_{X} K$. Then the tensor algebra of $ M$ is the free algebra on $ X$.
Proposition 2   $ (T(M),\iota')$ is a free associative algebra on $ X$.
Proof. Given any associative $ K$-algebra $ A$ and function $ f:X\rightarrow A$, then $ A$ is a $ K$-module and $ M$ is free on $ X$ so $ f$ extends to a unique $ K$-linear homomorphism $ \hat{f}:M\rightarrow A$.

Next we define $ K$-multilinear maps $ f^{(i)}:M^i\rightarrow A$ by

$\displaystyle f^{(i)}(m_1,\dots,m_i)=f(m_1)\cdots f(m_i).$
Then by the universal mapping property of tensor products (used inductively) we have a unique $ K$-linear map $ \hat{f}^{(i)}:T^i(M)\rightarrow A$ for which
$\displaystyle \hat{f}^{(i)}(m_1\otimes\cdots\otimes m_i)=f(m_1)\cdots f(m_i).$
Thus we have a unique algebra homomorphism $ \hat{f}^{(\infty)}:T(M)\rightarrow A$ such that $ \iota\hat{f}=f$. $ \qedsymbol$

This construction provides an obvious grading on the free algebra where the homogeneous components are

$\displaystyle T^n(M)=M^{\otimes n}=\bigotimes_{j=1}^n M.$

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\langle X\rangle$ and we will find many parallels to polynomial algebras with indeterminants in $ X$.

Let $ FM\langle X\rangle$ be the set of all words on $ X$. This makes $ FM\langle X\rangle$ a free monoid with identity the empty word and associative product the juxtaposition of words. Then define $ K\langle X\rangle$ as the $ K$-semi-group algebra on $ FM\langle X\rangle$. This means $ K\langle X\rangle$ is the free $ K$-modules oN $ FM\langle X\rangle$ and the product is defined as:

$\displaystyle \left(\sum_{w\in FM\langle X\rangle} l_w w\right) \left(\sum_{v\in FM\langle X\rangle} l_v v\right) =\sum_{w,v\in FM\langle X\rangle} l_v l_w wv.$

For example, $ \mathbb{Q}\langle x,y\rangle$ contains elements of the form

$\displaystyle x^2+4yxy,\qquad -7xy+2yx,\qquad 1+x+xy+xyx+x^2y+x^2y^2.$

This model of a free associative algebra encourages a mapping to polynomial rings. Indeed, $ K\langle X\rangle\rightarrow 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:

$\displaystyle x^2+4xy^2,\qquad -5xy,\qquad 1+x+xy+2x^2y+x^2y^2.$

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\langle X\rangle$ is homogeneous if it contained in $ FM\langle X\rangle$. 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\langle X\rangle$.

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.



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See Also: algebras, tensor algebra

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Cross-references: Jordan algebras, non-associative algebras, free modules, universal property, modules, component, span, length, homogeneous element, degree, contained, homogeneous, polynomial, translate, inclusion, property, polynomial rings, contains, juxtaposition, product, empty word, identity, free monoid, polynomial algebras, parallels, free groups, homogeneous components, grading, obvious, tensor products, free algebra, direct product, embedding, map, tensor, tensor algebra, basis, equivalent, naturally equivalent, objects, algebras, associative, category, commutative diagram, categorical, universal mapping property, homomorphism, algebra, functions, injection, unital ring, commutative, fix
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This is version 7 of free associative algebra, born on 2007-03-21, modified 2008-04-24.
Object id is 9098, canonical name is FreeAssociativeAlgebra.
Accessed 1356 times total.

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AMS MSC: 08B20 (General algebraic systems :: Varieties :: Free algebras)
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