q-Complex Numbers

• q-Complex Numbers are q-deformations of classical complex numbers characterized by modular invariance, noncommutative *-algebra structures, and continued fraction representations.
• Their analytic properties, including convergence radii and formal power series for q-irrationals, are established through methods like Rouché's theorem and q-continued fractions.
• Applications span quantum groups, knot invariants, and quantum topology, revealing deep ties with Chebyshev polynomials and q-deformed Gaussian integers.

The $q$-complex number is a $q$-deformation of classical complex numbers, developed to encode quantum, combinatorial, and modular symmetries. This notion arises from the program of $q$-deformation, where algebraic, analytic, and geometric objects are equipped with a formal parameter $q$ and inherit properties analogous to quantum deformations in representation theory, quantum groups, knot invariants, and enumerative combinatorics. Modern $q$-complex number theory is built on foundational constructions involving modular invariance, continued fraction deformations, noncommutative *-algebras, and analytic properties such as the radius of convergence in the complex $q$-plane. Recent advances have defined $q$-rationals and $q$-irrationals using modular group actions, produced explicit forms for $q$-deformed Gaussian integers, established convergence theorems, and articulated connections with Chebyshev polynomials and algebraic structures in quantum topology.

1. Foundational Constructions: $q$-Deformed Rational and Irrational Numbers

The theory starts with the classical $q$-analogue of integers: $[n]_q = 1 + q + q^2 + \cdots + q^{n-1}$, which generalizes to $q$-deformed rationals via an invariant map $[\,\cdot\,]_q:\mathbb{Q}\to\mathbb{Z}(q)$ designed to commute with the action of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ (Morier-Genoud et al., 31 Mar 2025). The generators $T$ and $S$ act on the classical projective line and their $q$-deformed analogues, $T_q(X) = qX+1$ and $S_q(X) = -1/(qX)$, satisfy $S_q^2 = (T_q S_q)^3 = 1$. For any rational $x=n/m$ in reduced form, its $q$-deformation is recursively computed either by modular action or by quantization of the continued fraction expansion: $$\left[\frac{n}{m}\right]_q = [c_0]_q - \frac{q^{c_0-1}}{ [c_1]_q - \frac{q^{c_1-1}}{ \cdots - \frac{q^{c_{\ell-1}-1}}{ [c_\ell]_q } } }$$ where $[n]_q$ are $q$-integers and $c_j$ are continued fraction coefficients (Leclere et al., 2021).

For irrationals, the stabilization phenomenon ensures that $q$-deformations converge coefficientwise for sequences of rational approximants, resulting in well-defined $q$-irrationals encoded by formal power series in $q$ (Morier-Genoud et al., 31 Mar 2025, Leclere et al., 2021). Explicitly, for irrational $x = [a_0; a_1, a_2, ...]$ with regular continued fraction, the $q$-irrational is represented by an infinite $q$-continued fraction generalizing the rational case.

2. Noncommutative *-Algebras and the Quantum Complex Plane

The algebraic structure of $q$-complex numbers is enriched by noncommutative *-algebra frameworks. A primary example is the algebra $\mathcal{A}$ generated by $x$ subject to the relation $xx^* = q x^* x$ for $q>0$, viewed as the coordinate algebra of the quantum complex plane (Cimpric et al., 2011). For $q=1$, this reduces to the commutative polynomial algebra, but for $q \neq 1$, the relation prescribes a $q$-deformation. Operators $X$ that satisfy $X X^* = q X^* X$ (or equivalently, $|X^* f| = q^{1/2} |X f|$ for all $f$ in the domain) are termed $q$-normal. A polar decomposition $X = U C$ yields further characterizations: $UCU^* = q^{1/2} C$ and $U E_C(A) U^* = E_C(q^{-1/2}A)$. The algebraic and operator-theoretic interplay forms a prototype for noncommutative real algebraic geometry in the $q$-deformed setting.

3. Modular Group Invariance and Quantized Complex Numbers

Central to the definition of $q$-complex numbers is the requirement of invariance under the modular group $\mathrm{PSL}(2,\mathbb{Z})$. For Gaussian integers $m+ni$ ($m,n\in \mathbb{Z}$), the $q$-deformed analogues are constructed via $[m+ni]_q = T_q^m U_q^n([0]_q)$, where $U_q$ is a $q$-deformed translation by $i$ compatible with the modular action (Ovsienko, 2021). The explicit forms of $q$-deformed Gaussian integers involve Euler's $q$-integers and auxiliary parameters such as $Q = (q^2 - q + 1)/(2i q^2 (q-1))$; for example,

$[2ni]_q = [2]_q [n]_Q [i]_q$

and similar for odd multiples. The translation operator $U_q$ requires nontrivial matrix constructions and parameter inversion, and its action produces rich combinatorial patterns.

The $q$-deformation for elliptic points, such as $i$, is determined by modular invariance: $[i]_q$ is defined as the unique fixed point of the $q$-deformed $S$ operator, ensuring compatibility throughout the construction.

4. Analytic Properties and Radius of Convergence

The analytic behavior of $q$-complex numbers, particularly $q$-irrationals, is characterized by their radius of convergence as a power series in $q$, viewed as a complex variable (Leclere et al., 2021, Ren, 2021). For $x \in \mathbb{R}$,

$R(x) = \limsup_n |\varkappa_n|^{1/n}$

where $[x]_q = \sum_{n} \varkappa_n q^n$. The $q$-deformed golden ratio $[\varphi]_q$ achieves the minimal radius $(3-\sqrt{5})/2 \approx 0.381966$ among all $q$-deformations, supported by computer experiments and partial results. For quadratic irrationals (the metallic numbers), $q$-continued fractions yield palindromic discriminants whose zeros set the radius bounds (Ren, 2021):

• For golden ratio: $R_* = (3-\sqrt{5})/2$
• For silver ratio: $R_1 \simeq 0.53101$
• For general metallic numbers: radii bounded below by $R_*$

Analytic lower bounds are proved using Rouché's theorem and palindromic properties of discriminants.

5. Algebraic and Combinatorial Structures

$q$-complex numbers inherit total positivity, unimodality, and palindromicity from their combinatorial origins. For rationals,

$\mathcal{X}_{n/m,\, n'/m'} = N(q)M'(q) - M(q)N'(q)$

is a polynomial with positive coefficients if $n/m > n'/m'$. This positivity extends through $q$-deformed Farey graphs and connections to lattice paths, Young diagrams, and snake graph enumerations.

Furthermore, $q$-deformed Gaussian integers exhibit a connection to Chebyshev polynomials of the second kind, arising from the recurrence satisfied by the imaginary parts of $[ni]_q$: $I_{n+2} = (Q+1)I_n - Q I_{n-2}$ mirroring the recurrence for $U_n(x)$, the Chebyshev polynomials (Ovsienko, 2021).

The $q$-deformed Heisenberg algebra provides a noncommutative context: the commutation relation $aa^+ - qa^+a = I$ creates a framework where compact operators correspond to derived Lie algebras and the Calkin algebra is isomorphic to $\mathbb{C}[x,x^{-1}]$, reflecting the algebraic structure of $q$-complex numbers (Cantuba, 2018).

6. Quantum Groups, Knot Invariants, and Topological Applications

The appearance of $q$-complex numbers is widespread in quantum groups, knot theory, and discrete geometry. Quantum groups utilize $q$-integers and $q$-binomials in their representation theory, while knot invariants—such as the Jones polynomial—frequently manifest $q$-deformed combinatorial objects (Morier-Genoud et al., 31 Mar 2025). In quantum topology, $q$-holonomic sequences and their transformation under twisting preserve $q$-holonomicity and underlie computations of colored Jones polynomials and the Kashaev invariant (Garoufalidis et al., 2012).

Topologically, $q$-deformed Aomoto complexes enable universal computations of local system cohomology for hyperplane arrangements, leveraging $q$-integers in the coboundary map matrices. The universality ensures that any specialization $q \to q_0 \in \mathbb{C}$ yields the correct local system (Yoshinaga, 1 Jan 2024).

7. Hidden Symmetries and Prospects for $q$-Complex Number Theory

Beyond PSL$(2,\mathbb{Z})$ invariance, the full symmetry group is $PGL(2,\mathbb{Z}) \times \mathbb{Z}_2$ acting on $q$-rational functions (Morier-Genoud et al., 31 Mar 2025). Natural involutions, such as $$\mathcal{I}(X(q)) = \frac{-X(q^{-1})+q-1}{(1-q)X(q^{-1})+q}$$, interchange "right" and "left" $q$-rational limits. For irrationals, the $q$-deformation is unique, but for rationals, bifurcation occurs, hinting at a two-component $q$-complex structure.

This symmetry may correspond to an extension of $q$-complex numbers, where the interplay between different $q$-rational limits supplies an analogue of real and imaginary parts, or more generally, a richer field-like structure. The extension and analysis of these symmetries is a subject of ongoing research.

Summary Table of Key $q$-Complex Number Constructions

Classical Object	$q$-Deformation/Construction	Main Structural Feature
Integer $n$	$[n]_q = 1 + q + \cdots + q^{n-1}$	Polynomial, unimodal coefficients
Rational $n/m$	$[n/m]_q$ via modular invariance/q-continued fraction	Rational function, positivity, palindromicity
Irrational $x$	$[x]_q$ by stabilization from rationals, infinite q-continued fraction	Formal power series, radius of convergence
Gaussian $m+n i$	$[m+ni]_q$ via $T_q^m U_q^n([0]_q)$	Modular invariance, Chebyshev polynomial link
Quantum complex plane	$\mathcal{A}$ with $xx^* = q x^* x$	Noncommutative *-algebra, $q$-normal operators
Heisenberg algebra	$a a^+ - q a^+ a = I$	Noncommutative, Calkin algebra $\cong \mathbb{C}[x,x^{-1}]$

These elements integrate modular group symmetries, combinatorial and operator-theoretic features, and analytic properties, establishing the conceptual and practical foundation of $q$-complex number theory (Cimpric et al., 2011, Ovsienko, 2021, Morier-Genoud et al., 31 Mar 2025, Leclere et al., 2021, Ren, 2021, Cantuba, 2018, Yoshinaga, 1 Jan 2024).

A plausible implication is that continued investigation of the symmetries inherent in $q$-deformed structures will produce a comprehensive theory analogous to classical complex analysis and number theory, with novel quantum, topological, and combinatorial phenomena.