$q$-deformed rationals and irrationals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908291841392640 |
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| author | Morier-Genoud, Sophie Ovsienko, Valentin |
| author_facet | Morier-Genoud, Sophie Ovsienko, Valentin |
| contents | The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial enumeration, discrete geometry, analysis, and many other parts of mathematics. In mathematical physics, $q$-deformations are often understood as ``quantizations''.
The recently introduced notion of a $q$-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 $q$-rational and a $q$-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of $q$-numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23834 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-deformed rationals and irrationals Morier-Genoud, Sophie Ovsienko, Valentin Combinatorics Quantum Algebra The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial enumeration, discrete geometry, analysis, and many other parts of mathematics. In mathematical physics, $q$-deformations are often understood as ``quantizations''. The recently introduced notion of a $q$-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 $q$-rational and a $q$-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of $q$-numbers. |
| title | $q$-deformed rationals and irrationals |
| topic | Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2503.23834 |