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

ocnmp:7480 - Open Communications in Nonlinear Mathematical Physics, July 29, 2021, Volume 1 - https://doi.org/10.46298/ocnmp.7480
Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

Authors: Valentin Ovsienko

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.

Volume: Volume 1
Published on: July 29, 2021
Accepted on: July 26, 2021
Submitted on: May 14, 2021
Keywords: Mathematics - Quantum Algebra,Mathematics - Combinatorics