Algebra and Combinatorics Seminar

Date: September 14, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Jurij Volcic, Texas A&M University

Title: Multipartite rational functions: the universal skew field of fractions of a tensor product of free algebras

Abstract: A commutative ring embeds into a field if and only if it has no zero divisors; moreover, in this case it admits a unique field of fractions. On the other hand, the problem of noncommutative localization and embeddings into skew fields (that is, division rings) is much more complex. For example, there exist rings without zero divisors that do not admit embeddings into skew fields, and rings with several non-isomorphic "skew fields of fractions". This led Paul Moritz Cohn (1924-2006) to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of two free algebras. After a decent introduction, we will look at the skew field of multipartite rational functions, whose construction via matrix evaluations of formal rational expressions is inspired by methods in free analysis. This skew field turns out to be the universal skew field of fractions of a tensor product of free algebras (for arbitrary finite number of factors).