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 \(ZP\)-basis (where \(P\) is the coefficient group) in quantum cluster algebras of finite and affine types. Key structure: quantum cluster variables satisfy \(X_i * X_j = q^{mᵢⱼ} X_j * X_i\) with \(mᵢⱼ\) 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 \(Ã_n, B̃_n, C̃_n, D̃_n, Ẽ_6, Ẽ_7, Ẽ_8, F̃_4, G̃_2\). These root systems encode the symmetries of lattices in 2, 3, 4 dimensions (e.g., \(G̃_2\) relates to hexagonal tiling, \(F̃_4\) to 24-cell). The quantum p Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com