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