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Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables

Lookup NU author(s): Professor Jim Agler, Professor Nicholas Young

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Abstract

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 𝕄^d. 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.

Publication metadata

Author(s): Agler J, McCarthy JE, Young NJ

Publication type: Article

Publication status: Published

Journal: Transactions of the London Mathematical Society

Year: 2018

Volume: 5

Issue: 1

Pages: 132–183

Print publication date: 01/12/2018

Online publication date: 14/11/2018

Acceptance date: 18/10/2008

Date deposited: 26/10/2018

ISSN (electronic): 2052-4986

Publisher: John Wiley & Sons Ltd.

URL: https://doi.org/10.1112/tlm3.12015

DOI: 10.1112/tlm3.12015

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