Multipartite Rational Functions

Abstract

Consider a tensor product of free algebras over a field $\mathbb{k}$, the so-called multipartite free algebra $\mathcal{A}=\mathbb{k}<X^{(1)}>\otimes \cdots \otimes \mathbb{k}<X^{(G)}>$. It is well-known that $\mathcal{A}$ is a domain, but not a fir nor even a Sylvester domain. Inspired by recent advances in free analysis, formal rational expressions over $\mathcal{A}$ together with their matrix representations in ${\mathrm{Mat}}_{n_1}(\mathbb{k})\otimes \cdots \otimes {\mathrm{Mat}}_{n_G}(\mathbb{k})$ are employed to construct a skew field of fractions $\mathcal{U}$ of $\mathcal{A}$, whose elements are called multipartite rational functions. It is shown that $\mathcal{U}$ is the universal skew field of fractions of $\mathcal{A}$ in the sense of Cohn. As a consequence a multipartite analog of Amitsur's theorem on rational identities relating evaluations in matrices over $\mathbb{k}$ to evaluations in skew fields is obtained. The characterization of $\mathcal{U}$ 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 $D\otimes \mathbb{k}<X>$ for an arbitrary skew field $D$ is given using matrix evaluations and formal rational expressions.