Free algebra

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra.

Definition

For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X₁,...,X_n} is the free R-module with a basis consisting of all words over the alphabet {X₁,...,X_n} (including the empty word, which is the unit of the free algebra). This R-module becomes an R-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words:

\left(X_{i_{1}}X_{i_{2}}\cdots X_{i_{l}}\right)\cdot \left(X_{j_{1}}X_{j_{2}}\cdots X_{j_{m}}\right)=X_{i_{1}}X_{i_{2}}\cdots X_{i_{l}}X_{j_{1}}X_{j_{2}}\cdots X_{j_{m}},

and the product of two arbitrary R-module elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear). This R-algebra is denoted R⟨X₁,...,X_n⟩. This construction can easily be generalized to an arbitrary set X of indeterminates.

In short, for an arbitrary set $X=\{X_{i}\,;\;i\in I\}$ , the free (associative, unital) R-algebra on X is

R\langle X\rangle :=\bigoplus _{w\in X^{\ast }}Rw

with the R-bilinear multiplication that is concatenation on words, where X* denotes the free monoid on X (i.e. words on the letters X_i), $\oplus$ denotes the external direct sum, and Rw denotes the free R-module on 1 element, the word w.

For example, in R⟨X₁,X₂,X₃,X₄⟩, for scalars α, β, γ, δ ∈ R, a concrete example of a product of two elements is

(\alpha X_{1}X_{2}^{2}+\beta X_{2}X_{3})\cdot (\gamma X_{2}X_{1}+\delta X_{1}^{4}X_{4})=\alpha \gamma X_{1}X_{2}^{3}X_{1}+\alpha \delta X_{1}X_{2}^{2}X_{1}^{4}X_{4}+\beta \gamma X_{2}X_{3}X_{2}X_{1}+\beta \delta X_{2}X_{3}X_{1}^{4}X_{4}

.

The non-commutative polynomial ring may be identified with the monoid ring over R of the free monoid of all finite words in the X_i.

Contrast with polynomials

Since the words over the alphabet {X₁, ...,X_n} form a basis of R⟨X₁,...,X_n⟩, it is clear that any element of R⟨X₁, ...,X_n⟩ can be written uniquely in the form:

\sum \limits _{k=0}^{\infty }\,\,\,\sum \limits _{i_{1},i_{2},\cdots ,i_{k}\in \left\lbrace 1,2,\cdots ,n\right\rbrace }a_{i_{1},i_{2},\cdots ,i_{k}}X_{i_{1}}X_{i_{2}}\cdots X_{i_{k}},

where $a_{i_{1},i_{2},...,i_{k}}$ are elements of R and all but finitely many of these elements are zero. This explains why the elements of R⟨X₁,...,X_n⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") X₁,...,X_n; the elements $a_{i_{1},i_{2},...,i_{k}}$ are said to be "coefficients" of these polynomials, and the R-algebra R⟨X₁,...,X_n⟩ is called the "non-commutative polynomial algebra over R in n indeterminates". Note that unlike in an actual polynomial ring, the variables do not commute. For example, X₁X₂ does not equal X₂X₁.

More generally, one can construct the free algebra R⟨E⟩ on any set E of generators. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z⟨E⟩.

Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. For a more general coefficient ring, the same construction works if we take the free module on n generators.

The construction of the free algebra on E is functorial in nature and satisfies an appropriate universal property. The free algebra functor is left adjoint to the forgetful functor from the category of R-algebras to the category of sets.

Free algebras over division rings are free ideal rings.

References

Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. Vol. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007.
L.A. Bokut' (2001) [1994], "Free associative algebra", Encyclopedia of Mathematics, EMS Press

Definition

Contrast with polynomials

See also

References