Free algebra explained

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 is the free R-module with a basis consisting of all words over the alphabet (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}} … X _{i _{l}} \right) ⋅ \left(X _{j _{1}} X _{j _{2}} … X _{j _{m}} \right) = X _{i _{1}} X _{i _{2}} … X _{i _{l}} X _{j _{1}} X _{j _{2}} … 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: = o plus _{w\in X ^{\ast}} R w 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), ⊕ 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 _{1X} ^{2} ^{2} + \beta X _{2X} _{3) ⋅ (\gamma} X _{2X} _{1} + \delta X ^{4 X} _{4 )} = \alpha\gamma X _{1X} ^{3 X} ^{1} + \alpha\delta X _{1X} ^{4 X} ^{4} + \beta\gamma X _{2X} _{3X} _{2X} _{1} + \beta\delta X _{2X} _{3X} ^{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 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 ^{inf ty} _{k =0} \sum\limits _{i _{1,i} _{2,} … ,i _{k\in\left\lbrace} 1 , 2 , … ,n\right\rbrace} a _{i _{1,i} _{2,} … ,i _{k}} X _{i _{1}} X _{i _{2}} … 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.

See also

• Cofree coalgebra
• Tensor algebra
• Free object
• Noncommutative ring
• Rational series
• Term algebra

References

• Book: Berstel . Jean . Reutenauer . Christophe . Noncommutative rational series with applications . Encyclopedia of Mathematics and Its Applications . 137 . Cambridge . . 2011 . 978-0-521-19022-0 . 1250.68007 .