No Primary Source for Quantum Parameter Inversion ↔ Affine Simple Reflection Correspondence — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The search returned no primary source establishing a "quantum parameter modular inversion q → q⁻¹ ↔ affine simple reflection s₀" theorem; results are adjacent (Q-systems, quantum affine/loop algebras, affine quantization, geometric Langlands) but none directly prove the claimed correspondence. | MATH: The relevant algebraic skeleton is: quantum parameter q ∈ ℂ*, inversion q ↦ q⁻¹; affine Weyl group generated by simple reflections s_i (i=0,…,n); Langlands dual Ǧ with root system Φ^∨; the affine reflection s₀ acts on the affine root lattice by w(α) = w(α) − ⟨α, α_i^∨⟩α_i. The q ↔ q⁻¹ symmetry is the standard *bar involution / Chevalley involution* on U_q(ĝ), not a new identity. | CONNECTION: Root systems (A_n, D_n, E_6,7,8) and affine extensions are crystallographic — their Cartan matrices have integer entries and admit the Coxeter/reflection structure. The q ↔ q⁻¹ map is an involution of the Dynkin diagram (a graph automorphism), which for simply-laced types is trivial, for B_n
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026primary,
title = {{No Primary Source for Quantum Parameter Inversion ↔ Affine Simple Reflection Correspondence — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658667},
url = {https://zenodo.org/records/23658667},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - No Primary Source for Quantum Parameter Inversion ↔ Affine Simple Reflection Correspondence — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658667
UR - https://zenodo.org/records/23658667
AB - FINDING: The search returned no primary source establishing a "quantum parameter modular inversion q → q⁻¹ ↔ affine simple reflection s₀" theorem; results are adjacent (Q-systems, quantum affine/loop algebras, affine quantization, geometric Langlands) but none directly prove the claimed correspondence. | MATH: The relevant algebraic skeleton is: quantum parameter q ∈ ℂ*, inversion q ↦ q⁻¹; affine Weyl group generated by simple reflections s_i (i=0,…,n); Langlands dual Ǧ with root system Φ^∨; the affine reflection s₀ acts on the affine root lattice by w(α) = w(α) − ⟨α, α_i^∨⟩α_i. The q ↔ q⁻¹ symmetry is the standard *bar involution / Chevalley involution* on U_q(ĝ), not a new identity. | CONNECTION: Root systems (A_n, D_n, E_6,7,8) and affine extensions are crystallographic — their Cartan matrices have integer entries and admit the Coxeter/reflection structure. The q ↔ q⁻¹ map is an involution of the Dynkin diagram (a graph automorphism), which for simply-laced types is trivial, for B_n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -