nLab noncommutative rational function

Idea

Rational function over an associative $k$-algebra $R$ should be a class of expressions involving operations in noncommutative polynomial ring combined with the operation of formal inverse modulo which are evaluable for at least at some choice of variables in $R$ in a chosen associative algebra modulo identities which are true after evaluation at common evaluation points.

Definitions

The division ring of noncommutative rational functions on a given alphabet is defined in Cohn’s book essentially as follows.

Noncommutative rational expressions

Given a commutative field $k$ (“field of constants”), and a finite set $X = \{x_1,\ldots,x_n\}$ (“alphabet”), consider the free algebra on $X$ (in the sense of universal algebra) of signature $\{ +_2, \cdot_2, (\,)^{-1}_1, -_1\}$ and with constants in $k$ and denote it by $\mathcal{R}(X,k)$. Expressions like $0^{-1}$ and $(x-x)^{-1}$ are in $\mathcal{R}(X,k)$ as no relations are imposed (one starts with terms which are elements of $X\cup k$ and continues by nesting sequence of algebraic operations eventually connecting all terms). The elements of $\mathcal{R}(X,k)$ are sometimes called noncommutative rational expressions on alphabet $X$.

Let $R$ be an associative $k$-algebra, and $\phi : X\to R$ a map of sets. Then there is a subset $\mathcal{R}_\phi \subset \mathcal{R}(X,k)$ (of $\phi$-“evaluables”) and a map $\phi_* : \mathcal{R}_\phi\to R$ uniquely determined by the rules

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

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

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

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

Domains

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

Equivalence of expressions

Two rational expressions with nonempty domains are equivalent if their domains have nonempty intersection and their evaluations agree at the intersection of domains, i.e. $f\sim g$ if $\forall\vec{r}\in Dom\,f\cap Dom\,g$ $\phi_{\vec{r}*}(f) = \phi_{\vec{r}*}(g)$.

Noncommutative rational functions

The equivalence classes of noncommutative rational expressions with nonempty domain are called noncommutative rational functions in variables in $X$ of $k$-algebra $R$. The domain of a rational function is the union of domains of all its representatives. If $k$ is an infinite field and under some conditions on $R$, noncommutative rational functions in alphabet $X$ over algebra $R$ naturally form a division ring (skewfield).

Free skewfield

If $R$ is the matrix ring $M_n(k)$ then all noncommutative rational functions in alphabet $X$ form a skewfield traditionally denoted by $k\lt\!\!\!\!\!(\,X\!\!\gt\!\!\!\!\!)$. It is the universal field of fractions of the noncommutative polynomial ring $k[X]$.

Amitsur considered instead $R = D$ where $D$ is any skewfield which is infinite-dimensional over its center $Z(D)$, where $Z(D)$ is infinite as well. He obtained the universal field of fractions which does not depend on $D$ in this construction either.

Properties

Cohn proved that if in the free skewfield any full matrix is invertible. An $n\times n$ matrix $A$ is full if it can not be factorized as $A_1 A_2$ where $A_1$ is $n\times r$, $A_2$ is $r\times n$ and $r\lt n$.

Literature

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.

The division ring of noncommutative rational functions on a given alphabet is defined in Cohn’s book.

• George M. Bergman, Skew fields of noncommutative rational functions, after Amitsur. In Séminaire Schützenberger–Lentin–Nivat, Année 1969/1970, No. 16. Paris 1970 EuDML
• P. M. Cohn, Universal skew fields of fractions, Symposia in Mathematics 8 (1972) 135–148
• C. Reutenauer, Inversion heights of free fields, Selecta Mathematica, N. S. 2:1 (1996) 93–109 pdf
• I. M. Gel'fand, V. S. Retakh, Determinants of matrices over noncommutative rings, Funkc. Anal. Pril. 25:2 (1991) 91–102; engl. transl. Funct. Anal. Appl. 21(1991) 51–58; A theory of noncommutative determinants and characteristic functions of graphs, Funkc. Anal. Pril. 26:4 (1992) 231–246.
• Zoran Škoda, Universal noncommutative flag variety, preprint (the exposition above is mainly taken from here)
• Daniel Krob, Expressions rationelles sur un anneau, in Marie-Paule Malliavin, Topics in invariant theory, Springer LNM 1478

Relation to free Lie algebras is in

• A. I. Lichtman, On universal fields of fractions for free algebras, J. Alg. 231 (2000) 652–676 doi

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

• Natalia Iyudu, Stanislav Shkarin, A proof of the Kontsevich periodicity conjecture, Duke Math. J. 164, no. 13 (2015) 2539–2575 doi arXiv:1305.1965

• M. Kontsevich, Noncommutative identities, writeup of the 2011 MPIM Bonn Arbeitstagung talk, arXiv:1109.2469

category: algebra