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Register a new userTowards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group
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Authors
Valentin Ovsienko
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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.
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We explain the notion of $q$-deformed real numbers introduced in our previous work and overview their main properties. We will also introduce $q$-deformed Conway-Coxeter friezes.
We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a $q$-analogue of a real. The construction is based on the notion of $q$-deformed rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.
We study analytic properties of $q$-deformed real numbers, a notion recently introduced by two of us. A $q$-deformed positive real number is a power series with integer coefficients in one formal variable~$q$. We study the radius of convergence of these power series assuming that $q in C.$ Our main conjecture, which can be viewed as a $q$-analogue of Hurwitzs Irrational Number Theorem, provides a lower bound for these radii, given by the radius of convergence of the $q$-deformed golden ratio. The conjecture is proved in several particular cases and confirmed by a number of computer experiments. For an interesting sequence of Pell polynomials, we obtain stronger bounds.
We develop further the theory of $q$-deformations of real numbers introduced by Morier-Genoud and Ovsienko, and focus in particular on the class of real quadratic irrationals. Our key tool is a $q$-deformation of the modular group $PSL_q(2,mathbb{Z})$. The action of the modular group by Mobius transformations commutes with the $q$-deformations. We prove that the traces of the elements of $PSL_q(2,mathbb{Z})$ are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the $q$-deformed quadratic irrationals.
Nonextensive statistical mechanics has been a source of investigation in mathematical structures such as deformed algebraic structures. In this work, we present some consequences of $q$-operations on the construction of $q$-numbers for all numerical sets. Based on such a construction, we present a new product that distributes over the $q$-sum. Finally, we present different patterns of $q$-Pascals triangles, based on $q$-sum, whose elements are $q$-numbers.
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