Rank-2 Root Systems and SU(2) Tensor Product Decomposition — E8 Intelligence Research

FINDING: Rank-2 root systems (A1×A1, A2, B2/C2, G2) are the complete classification of 2D crystallographic reflection groups; SU(2) tensor products decompose via A1 Cartan integers. | MATH: A1: single root α, Cartan matrix [2], Weyl group W(A1)≅S2 of order 2. SU(2)⊗SU(2): j1⊗j2 = ⊕_{j=|j1−j2|}^{j1+j2} j, dimension (2j1+1)(2j2+1)=Σ(2j+1). Cartan integer A_ij = 2(α_i·α_j)/(α_j·α_j); for A1×A1: diag(2,2); A2: [[2,−1],[−1,2]]; B2: [[2,−2],[−1,2]]; G2: [[2,−1],[−3,2]]. | CONNECTION: A2 root system = hexagonal lattice (60° angles, 6-fold symmetry) — direct base-60 / hexagonal crystallographic link. B2 = square lattice (90°). G2 = hexagonal with 30°/150° angles, 12 roots, order-12 Weyl group (dihedral D6). Weyl group orders: |W(A1×A1)|=4, |W(A2)|=6, |W(B2)|=8, |W(G2)|=12. Angle cosines between simple roots: A2: −1/2 (120°), B2: −√2/2 (135°), G2: −√3/2 (150°). | DEPTH: 7 FINDING: Cartan decomposition su(N) = 𝔨 ⊕ 𝔭 generalizes to arbitrary N via conjugate partitions and quotient algebras. | MA Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com