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

TL;DR

This paper introduces nc-manifolds as a non-commutative analogue of complex manifolds, illustrating their use in free analysis through non-commutative Riemann surfaces and symmetric functions, and deriving non-commutative Newton-Girard formulas.

Contribution

It defines nc-manifolds in several non-commuting variables and demonstrates their application in constructing non-commutative Riemann surfaces and symmetric functions, extending classical complex analysis concepts.

Findings

Construction of non-commutative Riemann surface for matrix square root

Development of non-commutative symmetric functions in two variables

Derivation of non-commutative Newton-Girard formulas

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 \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-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 $\m^2$ we construct a 2-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.