Noncommutative Riemann Surfaces

TL;DR

This paper develops noncommutative analogues of function algebras on Riemann surfaces using C*-algebras, analyzing their representations and matrix approximations, especially for surfaces interpolating between spheres and tori.

Contribution

It introduces C*-algebras for Riemann surfaces as noncommutative analogues and characterizes their finite-dimensional representations for a class of interpolating surfaces.

Findings

Sequences of matrix algebras approximate Poisson brackets as N→∞

Complete characterization of finite-dimensional representations for interpolating surfaces

Matrix-commutators converge to Poisson-brackets in the limit

Abstract

We introduce C-Algebras of compact Riemann surfaces $\Sigma$ as non-commutative analogues of the Poisson algebra of smooth functions on $\Sigma$. Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as $N\to\infty$. For a particular class of surfaces, nicely interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.