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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 ↗

Submitter琪越 唐

Version 1 / 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.

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. v1Initial depositCurrentSep 30, 2026