Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

TL;DR

This paper introduces a theory of $q$-deformed complex numbers, establishing their properties and symmetries, and relating them to Chebyshev polynomials, as a foundational step in quantum algebra.

Contribution

It proves the existence and uniqueness of a translation operator compatible with modular invariance in the context of $q$-deformed Gaussian integers.

Findings

Existence and uniqueness of translation operator by i

$q$-deformed Gaussian integers relate to Chebyshev polynomials

Invariance under modular group action

Abstract

This work is a first step towards a theory of "$q$-deformed complex numbers". Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials.