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 (_field of constants_), and a finite set , consider the free algebra on (in the sense of universal algebra) of signature and with constants in and denote it by . Expressions like and are in as no relations are imposed (one starts with terms which are elements of and continues by nesting sequence of algebraic operations eventually connecting all terms). Let be an associative -algebra, and a map of sets. Then there is a subset and a map uniquely determined by the rules
(constants evaluate) If then and .
(variables evaluate) If then and .
(sums, products and negatives of evaluables evaluate) If , then , , and .
If , and is invertible in , then and .
For every define to be the set of all -tuples such that where satisfies for . Those for which are called nondegenerate. It is clear that iff there is a subexpression in of the form where and is not invertible in .
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.
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, arxiv/1305.1965
M. Kontsevich, Noncommutative identities, writeup of the 2011 MPIM Bonn Arbeitstagung talk, arxiv/1109.2469
Last revised on May 11, 2013 at 22:09:53. See the history of this page for a list of all contributions to it.