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A bridge between analysis on local fields and quantum groups: the compact case

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath.NT — Number Theorymath.OA — Operator Algebrasmath.QA — Quantum Algebra

Contributed by Cédric ARHANCET ↗

Version 1 / Oct 29, 2026 / CC BY-SA 4.0

Abstract

Let KK be a non-Archimedean local field with residue cardinality QQ and let d≥1d \geq 1. We establish equivalence of Dirichlet forms between radial regional Taibleson operators on the unit ball of KdK^d and little qq-Jacobi operators. Under the parameter relation r2=Q−dr^2 = Q^{-d}, the zonal Casimir LL of the standard Podleś sphere yields equivalent forms through its powers Lα/dL^{\alpha/d} precisely for 0<α<3d0 < \alpha < 3d. Covariant quantum Markov realizations up to equivalence of forms exist precisely for 0<α≤d0 < \alpha \leq d. We also relate this correspondence to boundary energies: harmonic extensions on the tree of balls recover a Taibleson energy, while the energy per level of quantum Poisson integrals converges to a multiple of the Casimir energy. Finally, we extend the correspondence and the sharp covariance threshold to quantum projective spaces, identifying the Taibleson orders associated with their zonal Casimir operators.

Provenance statement

The main mathematical ideas and the overall research direction originated with the author. AI tools contributed to the development of some calculations and arguments. The author considers the manuscript broadly correct, but further detailed checking and several rounds of revision and polishing are still needed. This version is therefore shared as a work in progress, rather than as a fully checked final manuscript. The author takes responsibility for its content and welcomes corrections and comments.

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Version history

  1. v1Submitted by Cédric ARHANCETInitial depositCurrentOct 29, 2026