noncommutative rational function

Noncommutative rational identities are formally studied in

As explained in the Cohn’s book, one can define the division ring of noncommutative rational functions on a given alphabet.

Given a commutative field kk (_field of constants_), and a finite set X={x 1,,x n}X = \{x_1,\ldots,x_n\}, consider the free algebra on XX (in the sense of universal algebra) of signature {+ 2, 2,() 1 1, 1}\{ +_2, \cdot_2, (\,)^{-1}_1, -_1\} and with constants in kk and denote it by (X,k)\mathcal{R}(X,k). Expressions like 0 10^{-1} and (xx) 1(x-x)^{-1} are in (X,k)\mathcal{R}(X,k) as no relations are imposed (one starts with terms which are elements of XkX\cup k and continues by nesting sequence of algebraic operations eventually connecting all terms). Let RR be an associative kk-algebra, and ϕ:XR\phi : X\to R a map of sets. Then there is a subset ϕ(X,k)\mathcal{R}_\phi \subset \mathcal{R}(X,k) and a map ϕ *: ϕR\phi_* : \mathcal{R}_\phi\to R uniquely determined by the rules

For every f(X,k)f \in \mathcal{R}(X,k) define Dom ϕf\mathrm{Dom}_\phi\,f to be the set of all |X||X|-tuples r=(r 1,,r |X|)R |X|\vec{r} = (r_1,\ldots,r_{|X|}) \in R^{|X|} such that f ϕf \in \mathcal{R}_\phi where ϕ=ϕ r\phi = \phi_{\vec{r}} satisfies ϕ(x i)=r i\phi(x_i) = r_i for i=1,,|X|i = 1,\ldots, |X|. Those ff for which Dom ϕf\mathrm{Dom}_\phi f \neq \emptyset are called nondegenerate. It is clear that f ϕf \notin\mathcal{R}_\phi iff there is a subexpression in ff of the form (f 0) 1(f_0)^{-1} where f 0 ϕf_0 \in \mathcal{R}_\phi and ϕ *(f 0)\phi_*(f_0) is not invertible in RR.

It is used in the study of skewfields, Cohn localization, quasideterminants, noncommutative integrable systems and so on.

category: algebra