Langlands Duality: Killing Form Inversion and Quantum Weyl Action — E8 Intelligence Research

FINDING: Langlands duality maps root systems to coroot systems via inversion of the Killing form, preserving crystallographic lattice structure; recent work gives explicit quantum Weyl group action on coroot lattices. | MATH: For a simple Lie algebra \(g\), root system \(Φ\), coroot \(α^ = 2α/α,α\). Langlands dual \(^Lg\) has roots \(Φ^\). Killing form \(B\) induces isomorphism \(h → h^*\) via \(B(·, ·)\); inversion \(B⁻¹\) swaps long/short roots (ratio of squared lengths = 1, 2, or 3 for crystallographic systems). Quantum Weyl group action on coroot lattice \(Q^\) given by commuting generators \(e_x\) with \(x ∈ Q^\), formula: \(e_x = ∏ᵢ K_im_i · exp_q(...)\) (arXiv:2501.02365v2). | CONNECTION: Root systems are crystallographic — angles 60°, 90°, 120° (hexagonal, square, rectangular symmetries). Length ratios: \(A_n\): 1; \(B_n/C_n\): 1:√2 (≈1.414); Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com