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 \(a_{ij} = 2\langle \alpha_i, \alpha_j \rangle / \langle \alpha_i, \alpha_i \rangle\)), quantum cluster variables \(X_i\) with \(q\)-commutation relations \(X_i X_j = q^{d_{ij}} X_j X_i\) where \(d_{ij}\) is skew-symmetric from the exchange matrix; theta functions \(\vartheta_\gamma\) indexed by lattice points \(\gamma\) 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 \(h^\vee\); ratios like \(h^\vee/h\) 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