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Strongly positive bases in rank-two quantum cluster algebras

A mix of human-written and AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.RA — Rings and Algebras

Contributed by 琪越 唐

Version 2 / Sep 30, 2026 / CC BY 4.0

Abstract

We prove Conjecture 12 of Lee, Li, Rupel, and Zelevinsky: for arbitrary positive integers b,cb,c, every strongly positive basis of the coefficient-free rank-two quantum cluster algebra Av(b,c)\mathcal A_v(b,c) contains a unique element pointed at each pair in Z2\mathbb Z^2. These pointed elements exhaust the basis.

Provenance statement

The mathematical argument and manuscript were generated using OpenAI Codex (in just 15 minutes). The same AI system performed a separate adversarial review of the proof and exact finite checks of the normalization formulas. Uploading this paper to this platform does not mean that I claim ownership or copyright over it. My sole intention is to let more people know that the mathematical problem addressed in this paper has been solved by AI. Once it has been reviewed and its correctness has been verified, it may be cited directly. I welcome anyone who would like to organize this work; if you do so, the credit will be yours. Additionally, if you need more related literature or detailed materials, you can contact me by email. Email: tangqy24@mails.tsinghua.edu.cn

References

  1. A. Berenstein and A. Zelevinsky, Quantum cluster algebras , Adv. Math. 195 (2005), no. 2, 405--455. https://arxiv.org/abs/math/0404446 arXiv:math/0404446 .arXiv
  2. K. Lee, L. Li, D. Rupel, and A. Zelevinsky, Greedy bases in rank 2 quantum cluster algebras , Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9712--9716. https://arxiv.org/abs/1405.2311 arXiv:1405.2311 .arXiv

Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededSep 30, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionCurrentOct 04, 2026