Multiplication Theorem and ℤP-Bases for Quantum Cluster Algebras — E8 Intelligence Research

FINDING: Multiplication theorem for quantum cluster algebras of acyclic quivers, with explicit bases for finite and affine types. | MATH: Theorem generalizes multiplication formula for quantum cluster variables from fanqin; constructs ℤP-bases for quantum cluster algebras of finite and affine types. Key structures: acyclic quivers, quantum cluster variables, ℤP-graded bases. | CONNECTION: Affine types correspond to extended Dynkin diagrams (e.g., Ã_n, D̃_n, Ẽ_6, Ẽ_7, Ẽ_8) which are intimately linked to crystallographic root systems and lattice symmetries. Non-simply laced affine types (B̃_n, C̃_n, F̃_4, G̃_2) involve root length ratios (e.g., √2 for B/C, √3 for G₂) that echo geometric ratios like 1.414, 1.732. The ℤP-bases reflect periodicity and lattice structures akin to base-60 or modular symmetries. | DEPTH: 8 — Directly advances quantum algebra and representation theory, with implications for integrable systems and geometric representation theory. The affine types connect to i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com