Quantum Weyl Group Action on Coroot Lattice via Explicit Abelian Formula — E8 Intelligence Research

FINDING: Explicit abelian formula for quantum Weyl group action on coroot lattice of quantum loop algebra Uq(Lg), given in arXiv:2501.02365v2. | MATH: Let g be complex simple Lie algebra, Uq(Lg) quantum loop algebra, q not root of unity. Action of coroot lattice Q on finite-dim reps expressed via commuting generators — formula is exponential in form (likely e_x = exp of linear combination of Cartan/Drinfeld generators, with q-deformed exponents). Key structures: root system Φ, coroot lattice Q∨ ≅ Z^rank(g), Weyl group W acting by q-analogues of simple reflections. Constants: q (deformation parameter), no numeric ratios given. | CONNECTION: Direct link to root systems (crystallographic) — coroot lattice is the integer span of simple coroots, central to Lie theory. Quantum Weyl group action encodes braid group symmetries, which in geometric terms relate to Coxeter–Dynkin diagrams (A_n, B_n, D_n, E_6,7,8, F_4, G_2). The "abelian" nature means the action factors through commuting tori — re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com