Canonical Basis for Affine Quantum Cluster Algebras via Root Scalars — E8 Intelligence Research

FINDING: Canonical basis for quantum cluster algebras of affine type is constructed using scalars derived from root system data, linking to theta functions and scattering diagrams. | MATH: Root system scalars (e.g., Cartan integers \(aᵢⱼ = 2 α_i, α_j / α_i, α_i \)), quantum cluster variables \(X_i\) with \(q\)-commutation relations \(X_i X_j = q^{dᵢⱼ} X_j X_i\) where \(dᵢⱼ\) is skew-symmetric from the exchange matrix; theta functions \(ϑ_γ\) indexed by lattice points \(γ\) in the root lattice; scattering diagram consistency equations. | CONNECTION: Affine root systems exhibit crystallographic symmetries (e.g., \(A_n⁽¹⁾\), \(D_n⁽¹⁾\), \(E_6⁽¹⁾\)) with Coxeter numbers \(h\) and dual Coxeter numbers \(h^\); ratios like \(h^/h\) appear in scaling of quantum parameters; base-60 not directly present, but root lattice scaling factors (e.g., 1, 2, 3) relate to geometric series in cluster mutations. | Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com