Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symmetries of the q-deformed real projective line

Perrine Jouteur

TL;DR

By quantizing the modular group's action on the real projective line in two steps, the paper extends Morier-Genoud–Ovsienko's $q$-deformation from $PSL_2( olinebreak \\mathbb{Z})$ to $PGL_2( olinebreak \\mathbb{Z})$ and, further, to a $\mathbb{Z}_2$-twisted duplication, providing left/right $q$-rational numbers and a robust framework for the quantization of algebraic relations. The main methods include defining $N_q$ and $I_q$ to mimic determinant $-1$ and an involution, introducing $\tau$, $\bar I_q$, and twisted actions, and proving the injectivity of the quantization maps for irrational inputs. The paper obtains palindromic positivity properties of $q$-traces, develops $q$-continued fraction machinery, and applies the framework to algebraic numbers of degrees $4$ and $6$, deriving quantized Vieta relations and analyzing Galois actions via $q$-deformed linear fractional transformations. Overall, it establishes a coherent, extendable $q$-symmetry structure for $q$-deformed real numbers with concrete algebraic applications.

Abstract

We generalize in two steps the quantized action of the modular group on $q$-deformed real numbers introduced by Morier-Genoud and Ovsienko. First, we let the projective general linear group $PGL_2(\mathbb{Z})$ act on $q$-real numbers via a $q$-deformed action. The quantized matrices we get have combinatorial interpretations. Then we consider an extension of the group $PGL_2(\mathbb{Z})$ by the $2$-elements cyclic group, and define a quantized action of this extension on $q$-real numbers. We deduce from these actions some underlying relations between $q$-real numbers, and between left and right versions of $q$-deformed rational numbers. In particular we investigate the case of some algebraic numbers of degree $4$ and $6$. We also prove that the way of quantizing real numbers defined by Morier-Genoud and Ovsienko is an injective process.

Symmetries of the q-deformed real projective line

TL;DR

By quantizing the modular group's action on the real projective line in two steps, the paper extends Morier-Genoud–Ovsienko's -deformation from to and, further, to a -twisted duplication, providing left/right -rational numbers and a robust framework for the quantization of algebraic relations. The main methods include defining and to mimic determinant and an involution, introducing , , and twisted actions, and proving the injectivity of the quantization maps for irrational inputs. The paper obtains palindromic positivity properties of -traces, develops -continued fraction machinery, and applies the framework to algebraic numbers of degrees and , deriving quantized Vieta relations and analyzing Galois actions via -deformed linear fractional transformations. Overall, it establishes a coherent, extendable -symmetry structure for -deformed real numbers with concrete algebraic applications.

Abstract

We generalize in two steps the quantized action of the modular group on -deformed real numbers introduced by Morier-Genoud and Ovsienko. First, we let the projective general linear group act on -real numbers via a -deformed action. The quantized matrices we get have combinatorial interpretations. Then we consider an extension of the group by the -elements cyclic group, and define a quantized action of this extension on -real numbers. We deduce from these actions some underlying relations between -real numbers, and between left and right versions of -deformed rational numbers. In particular we investigate the case of some algebraic numbers of degree and . We also prove that the way of quantizing real numbers defined by Morier-Genoud and Ovsienko is an injective process.

Paper Structure

This paper contains 21 sections, 28 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Background on $q$-deformed numbers
  3. First extension of symmetries
  4. Second extension of symmetries
  5. Application to irrational algebraic numbers
  6. Quantized action of the projective linear group
  7. From $\mathop{\mathrm{PSL}}\nolimits_2(\mathbb{Z})$ to $\mathop{\mathrm{PGL}}\nolimits_2(\mathbb{Z})$
  8. Quantized positive continued fractions
  9. Palindromicity and positivity of $q$-traces in determinant $-1$
  10. Duplication of $\mathop{\mathrm{PGL}}\nolimits_2(\mathbb{Z})$
  11. Twisted matrices
  12. Action on rational numbers
  13. Focus on $q$-real numbers
  14. Convergences
  15. Injectivity of quantization
  16. ...and 6 more sections

Key Result

Proposition 1.1

There are only two $q$-deformations of $\infty$ in $\mathbb{P}^1(\mathbb{Z}(q))$ whose orbits under the action of $\mathop{\mathrm{PSL}}\nolimits_{2,q}(\mathbb{Z})$ satisfy the condition : These two $q$-deformations of $\infty$ are

Figures (2)

  • Figure 1: Galois group action on the roots by linear fractional transformations
  • Figure 2: Galois group action on the roots by linear fractional transformations

Theorems & Definitions (64)

  • Proposition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 54 more