Rational function over an associative -algebra 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 in a chosen associative algebra modulo identities which are true after evaluation at common evaluation points.
The division ring of noncommutative rational functions on a given alphabet is defined in Cohn’s book essentially as follows.
Given a commutative field (“field of constants”), and a finite set (“alphabet”), 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). The elements of are sometimes called noncommutative rational expressions on alphabet .
Let be an associative -algebra, and a map of sets. Then there is a subset (of -“evaluables”) 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 .
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. if .
The equivalence classes of noncommutative rational expressions with nonempty domain are called noncommutative rational functions in variables in of -algebra . The domain of a rational function is the union of domains of all its representatives. If is an infinite field and under some conditions on , noncommutative rational functions in alphabet over algebra naturally form a division ring (skewfield).
If is the matrix ring then all noncommutative rational functions in alphabet form a skewfield traditionally denoted by . It is the universal field of fractions of the noncommutative polynomial ring .
Amitsur considered instead where is any skewfield which is infinite-dimensional over its center , where is infinite as well. He obtained the universal field of fractions which does not depend on in this construction either.
Cohn proved that if in the free skewfield any full matrix is invertible. An matrix is full if it can not be factorized as where is , is and .
Noncommutative rational identities are formally studied in
The division ring of noncommutative rational functions on a given alphabet is defined in Cohn’s book.
Relation to free Lie algebras is in
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
Last revised on July 19, 2024 at 17:28:21. See the history of this page for a list of all contributions to it.