noncommutative rational function

Noncommutative rational identities are formally studied in

  • S. A. Amitsur, Rational identities and application to algebra and geometry, J. Alg. 3 (1966), 304–359.
  • P. M. Cohn, Free rings and their relations, Academic Press 1971.

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

  • (constants evaluate) If ckc \in k then c ϕc \in \mathcal{R}_\phi and ϕ *(c)=c\phi_*(c) = c.

  • (variables evaluate) If xXx \in X then x ϕx \in \mathcal{R}_\phi and ϕ *(x)=ϕ(x)\phi_*(x) = \phi(x).

  • (sums, products and negatives of evaluables evaluate) If f,g ϕf, g \in\mathcal{R}_\phi, then f+g,fg,f ϕf+g, f\cdot g, -f \in\mathcal{R}_\phi, ϕ *(f+g)=ϕ *(f)+ϕ *(g)\phi_*(f+g) = \phi_*(f) + \phi_*(g), ϕ *(fg)=ϕ *(f)ϕ *(g)\phi_*(f\cdot g) = \phi_*(f)\cdot\phi_*(g) and ϕ *(f)=ϕ *(f)\phi_*(-f) = -\phi_*(f).

  • If g ϕg \in \mathcal{R}_\phi, and ϕ *(g)\phi_*(g) is invertible in RR, then g 1 ϕg^{-1}\in \mathcal{R}_\phi and ϕ *(g 1)=ϕ *(g) 1\phi_*(g^{-1}) = \phi_*(g)^{-1}.

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.

  • I. M. Gel'fand, V. S. Retakh, Determinants

    of matrices over noncommutative rings_, Funct. Anal. Appl. 25 (1991), no.2, pp. 91–102; engl. transl. 21 (1991), pp. 51–58; A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992), no.4, pp. 231–246.

  • Zoran Škoda, Universal noncommutative flag variety, preprint (the exposition above is mainly taken from here)

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

category: algebra

Last revised on May 11, 2013 at 22:09:53. See the history of this page for a list of all contributions to it.