$q$-deformed rationals and irrationals
Abstract
The concept of -deformation, or ``-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; -deformations are important for knot invariants, combinatorial enumeration, discrete geometry, analysis, and many other parts of mathematics. In mathematical physics, -deformations are often understood as ``quantizations''. The recently introduced notion of a -deformed real number is based on the geometric idea of invariance by a modular group action. The goal of this lecture is to explain what is a -rational and a -irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of -numbers.
Cite
@article{arxiv.2503.23834,
title = {$q$-deformed rationals and irrationals},
author = {Sophie Morier-Genoud and Valentin Ovsienko},
journal= {arXiv preprint arXiv:2503.23834},
year = {2025}
}
Comments
This lecture is a contribution to the second edition of the book "Mathematical omnibus", American Mathematical Society, by Dmitry Fuchs, and Serge Tabachnikov. The exposition is accessible for undergraduate students