q-deformations of the modular group and of the real quadratic\n irrational numbers

2021/01/08 by Ludivine Leclere, Leclere, Ludivine, Sophie Morier-Genoud +1 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2101.02953

openalex publication_date 2021/01/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We develop further the theory of q-deformations of real numbers introduced\nby Morier-Genoud and Ovsienko, and focus in particular on the class of real\nquadratic irrationals. Our key tool is a q-deformation of the modular group\nPSL_q(2,\ℤ). The action of the modular group by M "obius\ntransformations commutes with the q-deformations. We prove that the traces of\nthe elements of PSL_q(2,\ℤ) are palindromic polynomials with positive\ncoefficients. These traces appear in the explicit expressions of the\nq-deformed quadratic irrationals.\n

Cited by

• q-rationals and dimers
• On q-real and q-complex numbers
• On q-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges

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