Agler, J, McCarthy, JE and Young, NJ (2018) Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables. Transactions of the London Mathematical Society, 5 (1). pp. 132-183. ISSN 2052-4986
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 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 2 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2018 The Authors. The Transactions of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | 15A54; 32A99; 58A05; 58J42 (primary) |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number Newcastle University/EPSRC BH 122321 |
Depositing User: | Symplectic Publications |
Date Deposited: | 22 Oct 2018 09:08 |
Last Modified: | 25 Jun 2023 21:33 |
Status: | Published |
Publisher: | London Mathematical Society |
Identification Number: | 10.1112/tlm3.12015 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:137461 |
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