TY - JOUR

T1 - Multipartite Rational Functions

AU - Klep, Igor

AU - Vinnikov, Victor

AU - Volˇciˇ, Jurij

N1 - Funding Information: The first author was supported by the Slovenian Research Agency grants J1-8132, J1-2453 and P1-0222 and partially supported by the Marsden Fund Council of the Royal Society of New Zealand. The second author was supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1. The third author was supported by the NSF grant DMS 1954709 and partially supported by the University of Auckland Doctoral Scholarship and by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1. The authors thank Roland Speicher for drawing the free probability aspect of multipartite rational functions to their attention. Publisher Copyright: © 2020, Documenta Mathematica. All Rights Reserved.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - Consider a tensor product of free algebras over a field k, the so-called multipartite free algebra A = k<X(1)>⊗· · · ⊗ k<X(G)>. It is well-known that A is a domain, but not a fir nor even a Sylvester domain. Inspired by recent advances in free analysis, formal rational expressions over A together with their matrix representations in Matn1 (k)⊗· · ·⊗MatnG(k) are employed to construct a skew field of fractions U of A, whose elements are called multipartite rational functions. It is shown that U is the universal skew field of fractions of A in the sense of Cohn. As a consequence a multipartite analog of Amitsur’s theorem on rational identities relating evaluations in matrices over k to evaluations in skew fields is obtained. The characterization of U in terms of matrix evaluations fits naturally into the wider context of free noncommutative function theory, where multipartite rational functions are interpreted as higher order noncommutative rational functions with an associated difference-differential calculus and linear realization theory. Along the way an explicit construction of the universal skew field of fractions of D ⊗ k<X> for an arbitrary skew field D is given using matrix evaluations and formal rational expressions.

AB - Consider a tensor product of free algebras over a field k, the so-called multipartite free algebra A = k<X(1)>⊗· · · ⊗ k<X(G)>. It is well-known that A is a domain, but not a fir nor even a Sylvester domain. Inspired by recent advances in free analysis, formal rational expressions over A together with their matrix representations in Matn1 (k)⊗· · ·⊗MatnG(k) are employed to construct a skew field of fractions U of A, whose elements are called multipartite rational functions. It is shown that U is the universal skew field of fractions of A in the sense of Cohn. As a consequence a multipartite analog of Amitsur’s theorem on rational identities relating evaluations in matrices over k to evaluations in skew fields is obtained. The characterization of U in terms of matrix evaluations fits naturally into the wider context of free noncommutative function theory, where multipartite rational functions are interpreted as higher order noncommutative rational functions with an associated difference-differential calculus and linear realization theory. Along the way an explicit construction of the universal skew field of fractions of D ⊗ k<X> for an arbitrary skew field D is given using matrix evaluations and formal rational expressions.

KW - Universal skew field of fractions

KW - free function theory

KW - free skew field

KW - multipartite rational function

KW - noncommutative rational function

KW - tensor product of free algebras

UR - http://www.scopus.com/inward/record.url?scp=85126713281&partnerID=8YFLogxK

U2 - https://doi.org/10.4171/DM/777

DO - https://doi.org/10.4171/DM/777

M3 - Article

SN - 1431-0635

VL - 25

SP - 1285

EP - 1314

JO - Documenta Mathematica

JF - Documenta Mathematica

ER -