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Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group (2103.10800v4)
Published 18 Mar 2021 in math.QA and math.CO
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.
