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 \\(\\mathfrak{g}\\), root system \\(\\Phi\\), coroot \\(\\alpha^\\vee = 2\\alpha/\\langle\\alpha,\\alpha\\rangle\\). Langlands dual \\(^L\\mathfrak{g}\\) has roots \\(\\Phi^\\vee\\). Killing form \\(B\\) induces isomorphism \\(\\mathfrak{h} \\to \\mathfrak{h}^*\\) via \\(B(\\cdot, \\cdot)\\); 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^\\vee\\) given by commuting generators \\(e_x\\) with \\(x \\in Q^\\vee\\), formula: \\(e_x = \\prod_{i} K_i^{m_i} \\cdot \\exp_q(\\text{...})\\) (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