A New Class of Two-Dimensional Noncommutative Spaces

TL;DR

This paper introduces a new class of two-dimensional noncommutative geometries with differential structures, generalizing the noncommutative plane, and demonstrates how to recover classical punctured planes through continuum limits using generalized coherent states.

Contribution

It presents an infinite family of noncommutative geometries with differential structures, extending the two-dimensional noncommutative plane and enabling continuum limits via generalized coherent states.

Findings

Infinite noncommutative geometries with differential structures identified

Continuum limit recovers punctured plane with non-constant Poisson structures

Infinite dimensional representations of these geometries established

Abstract

We find an infinite number of noncommutative geometries which posses a differential structure. They generalize the two dimensional noncommutative plane, and have infinite dimensional representations. Upon applying generalized coherent states we are able to take the continuum limit, where we recover the punctured plane with non constant Poisson structures.