Algebra of Noncommutative Riemann Surfaces

TL;DR

This paper explores the algebraic structures of noncommutative Riemann surfaces, developing a framework for noncommutative complex analysis and examining their geometric and functional properties.

Contribution

It introduces a comprehensive algebraic framework for noncommutative Riemann surfaces derived from noncommutative field theory and extends complex analysis to this setting.

Findings

Established a noncommutative complex coordinate system

Analyzed conformal mappings in noncommutative context

Explored Teichmüller theory for noncommutative surfaces

Abstract

We examine several algebraic properties of the noncommutive $z$-plane and Riemann surfaces. The starting point of our investigation is a two-dimensional noncommutative field theory, and the framework of the theory will be converted into that of a complex coordinate system. The basis of noncommutative complex analysis is obtained thoroughly, and the considerations on functional analysis are also given before performing the examination of the conformal mapping and the Teichm\"{u}ller theory. (Keywords; Complex Analysis, Riemann Surfaces and Teichm\"{u}ller Space, Functional Analysis, Deformation Quantization, Non-Commutative Geometry, Quantum Groups)