Noncommutative rational functions and their finite-dimensional representations

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dc.contributor.advisor Klep, I en Volcic, Jurij en 2018-01-14T22:51:53Z en 2017 en
dc.identifier.uri en
dc.description.abstract Noncommutative rational functions form a free skew field, a universal object in the category of skew fields. They emerge in various branches of pure and applied mathematics, such as noncom-mutative algebra, automata theory, control theory, free analysis, free real algebraic geometry and free probability. A noncommutative rational function is given by a formal rational expression involving freely noncommuting variables and arithmetic operations, which can be naturally evaluated at tuples of matrices of all sizes. This dissertation studies such finite-dimensional evaluations, with the goal of determining what information about the structure of the free skew field can be recovered from them. Firstly a matrix coefficient realization theory is developed, which for an arbitrary noncom-mutative rational function yields an efficiently computable normal form. Such a realization of a noncommutative rational function measures its complexity and describes its natural domain as the complement of a free singularity locus of a linear matrix pencil. The inclusion problem for free loci, and thus for domains of noncommutative rational functions, is next solved in terms of epimorphisms between the coefficient algebras of the corresponding linear matrix pencils. This theorem, called a Singularitatstellensatz for linear matrix pencils, is closely related to the invariant theory of the general linear group. Via realization theory this result yields an algebraic characterization of noncommutative rational functions with a given domain. More-over, a description of noncommutative rational functions whose domains contain all tuples of hermitian matrices is derived. In particular, it is proven that an everywhere defined noncom-mutative rational function is a noncommutative polynomial. The understanding of the behavior of noncommutative rational functions under hermitian evaluations is further advanced by the resolution of an noncommutative analog of Hilbert’s 17th problem: a noncommutative rational function that is positive semidefinite at every tuple of hermitian matrices is a sum of hermitian squares of noncommutative rational functions. Finally, the construction of the free skew field via matrix evaluations is extended to partially commuting arguments. The obtained multipartite rational functions play a remarkable role in the theory of universal skew fields of fractions and in the difference-differential calculus in free analysis. en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA99264960113702091 en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
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dc.rights.uri en
dc.title Noncommutative rational functions and their finite-dimensional representations en
dc.type Thesis en Mathematics en The University of Auckland en Doctoral en PhD en
dc.rights.holder Copyright: The author en
dc.rights.accessrights en
pubs.elements-id 720943 en
pubs.record-created-at-source-date 2018-01-15 en
dc.identifier.wikidata Q112201062

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