## 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 (Formula presented.) -dimensional nc-universe (Formula presented.). 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 (Formula presented.) 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.

Original language | English |
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Pages (from-to) | 132-183 |

Number of pages | 52 |

Journal | Transactions of the London Mathematical Society |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2018 |

## Keywords

- 15A54
- 32A99
- 58A05
- 58J42 (primary)