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

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.

21 pages

Cited by in corpus (2)

• A $q$-analog of the Markoff injectivity conjecture holds
• Infinitesimal Modular Group: $q$-Deformed ${\mathfrak{sl}}_2$ and Witt Algebra