Lookup NU author(s): Professor Jim Agler,
Professor Nicholas Young
This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).
The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of nc-manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc-domain in the d-dimensional nc-universe 𝕄d. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in 𝕄2 we construct a two-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.
Author(s): Agler J, McCarthy JE, Young NJ
Publication type: Article
Publication status: Published
Journal: Transactions of the London Mathematical Society
Print publication date: 01/12/2018
Online publication date: 14/11/2018
Acceptance date: 18/10/2008
ISSN (electronic): 2052-4986
Publisher: John Wiley & Sons Ltd.
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