Abelian Formula for Quantum Weyl Group Action on Loop Modules — E8 Intelligence Research

FINDING: Explicit abelian formula for quantum Weyl group action of coroot lattice on finite-dimensional \\(U_q(L\\mathfrak{g})\\)-modules via commuting generators. | MATH: Let \\(\\mathfrak{g}\\) be complex simple, \\(U_q(L\\mathfrak{g})\\) quantum loop algebra, \\(q\\) not root of unity. Coroot lattice \\(Q = \\bigoplus_i \\mathbb{Z}\\alpha_i^\\vee\\). Action expressed as product over simple roots: \\(\\mathcal{S}_{\\alpha_i^\\vee} = \\exp_q\\left( \\frac{1}{(q-q^{-1})} \\cdot \\text{log}_q(\\text{commuting generator}_i) \\right)\\) — the paper (arXiv:2501.02365v2) gives closed form in terms of the Drinfeld–Jimbo commuting generators \\(\\{k_i^\\pm, h_{i,n}\\}\\). Key structure: the Weyl group action factorizes into commuting parts, breaking the usual braid group complexity into abelian exponentials. | CONNECTION: Coroot lattice \\(Q\\) is a root lattice — crystallographic, integral, with Cartan matrix symmetries. For \\(\\mathfrak{g}=A_n\\), the coroot lattice is the \\(A_n\\) lattice, whose Voronoi cell and kissing number Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com