Quantum Weyl Group Actions on Coroot Lattices via Langlands Duality — E8 Intelligence Research

FINDING: Langlands duality maps root systems to coroot systems via inversion, with quantum Weyl group actions on coroot lattices now explicitly formulated. | MATH: Root system Φ ⊂ V, coroot system Φ∨ ⊂ V*; Killing form B induces isomorphism V ≅ V* via x ↦ B(x,·), sending α ↦ 2α/B(α,α) = α∨. Weyl group W acts on both, with w(α∨) = (wα)∨. Quantum loop algebra Uq(L𝔤) action on coroot lattice Q: explicit abelian formula for quantum Weyl group action, ex = product over commuting generators (arXiv:2501.02365v2). | CONNECTION: Root systems are crystallographic by definition (2B(α,β)/B(α,β) ∈ ℤ). The inversion α ↔ α∨ is a duality that preserves the Cartan matrix up to transpose — for simply-laced types (A,D,E) this is identity, but for B_n ↔ C_n it swaps long/short roots, giving ratio √2 between root lengths. This is the same √2 that appears in crystallographic lattice constants and in the 45° rotation symmetry of the square lattice. The Killing form normalization yields the dual Coxeter numbe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com