Multiplication Theorem and Basis Construction for Acyclic Quantum Cluster Algebras — E8 Intelligence Research

FINDING: Multiplication theorem for quantum cluster algebras of acyclic quivers, with explicit basis construction for finite and affine types. MATH: The theorem generalizes the multiplication formula for quantum cluster variables from [fanqin]. It yields a \(\mathbb{ZP}\)-basis (where \(\mathbb{P}\) is the coefficient group) in quantum cluster algebras of finite and affine types. Key structure: quantum cluster variables satisfy \(X_i * X_j = q^{m_{ij}} X_j * X_i\) with \(m_{ij}\) from the exchange matrix. The basis is built from monomials in cluster variables with quantum corrections. CONNECTION: Affine types correspond to extended Dynkin diagrams, which are crystallographic root systems of types \(\tilde{A}_n, \tilde{B}_n, \tilde{C}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8, \tilde{F}_4, \tilde{G}_2\). These root systems encode the symmetries of lattices in 2, 3, 4 dimensions (e.g., \(\tilde{G}_2\) relates to hexagonal tiling, \(\tilde{F}_4\) to 24-cell). The quantum p Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com