A₁ Root System: Weyl Group S₂ Underlying SU(2) and Topological Phases — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: A₁ root system is the minimal rank-1 case whose Weyl group is S₂ = ℤ₂; its representation theory (Clebsch–Gordan / 3j-symbols) is the algebraic skeleton underlying SU(2) angular momentum and, via Altland–Zirnbauer classes, topological phase classification.
MATH: - A₁ root system: roots {±α}, Cartan matrix [2], Weyl group W(A₁) = S₂ = {1, s_α}, s_α(α) = −α. Rank 1, |W| = 2. - Clebsch–Gordan: V_{j₁} ⊗ V_{j₂} = ⊕_{J=|j₁−j₂|}^{j₁+j₂} V_J. For A₁ (SU(2)), multiplicity-free. - Wigner 3j-symbol relates to CG: ⟨j₁m₁j₂m₂|JM⟩ = (−1)^{j₁−j₂+M} √(2J+1) · (j₁ j₂ J; m₁ m₂ −M). - Selection rules (triangular): |j₁−j₂| ≤ J ≤ j₁+j₂; m₁+m₂+M = 0. - Orthogonality: Σ_{m₁m₂} (j₁ j₂ j₃; m₁ m₂ m₃)(j₁ j₂ j₃'; m₁ m₂ m₃') = δ_{j₃j₃'}δ_{m₃m₃'} Δ(j₁j₂j₃), where Δ is the triangle coefficient. - Symmetry: 3j invariant under even permutations of columns; odd permutations give phase (−1)^{j₁+j₂+j₃}. - AZ classification: 2 complex (A, AIII) + 8 real (AI, BDI, D, DIII, AII, CII, C, CI) classes ↔ symmetric spac
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026a₁,
title = {{A₁ Root System: Weyl Group S₂ Underlying SU(2) and Topological Phases — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658881},
url = {https://zenodo.org/records/23658881},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A₁ Root System: Weyl Group S₂ Underlying SU(2) and Topological Phases — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658881
UR - https://zenodo.org/records/23658881
AB - FINDING: A₁ root system is the minimal rank-1 case whose Weyl group is S₂ = ℤ₂; its representation theory (Clebsch–Gordan / 3j-symbols) is the algebraic skeleton underlying SU(2) angular momentum and, via Altland–Zirnbauer classes, topological phase classification. MATH: - A₁ root system: roots {±α}, Cartan matrix [2], Weyl group W(A₁) = S₂ = {1, s_α}, s_α(α) = −α. Rank 1, |W| = 2. - Clebsch–Gordan: V_{j₁} ⊗ V_{j₂} = ⊕_{J=|j₁−j₂|}^{j₁+j₂} V_J. For A₁ (SU(2)), multiplicity-free. - Wigner 3j-symbol relates to CG: ⟨j₁m₁j₂m₂|JM⟩ = (−1)^{j₁−j₂+M} √(2J+1) · (j₁ j₂ J; m₁ m₂ −M). - Selection rules (triangular): |j₁−j₂| ≤ J ≤ j₁+j₂; m₁+m₂+M = 0. - Orthogonality: Σ_{m₁m₂} (j₁ j₂ j₃; m₁ m₂ m₃)(j₁ j₂ j₃'; m₁ m₂ m₃') = δ_{j₃j₃'}δ_{m₃m₃'} Δ(j₁j₂j₃), where Δ is the triangle coefficient. - Symmetry: 3j invariant under even permutations of columns; odd permutations give phase (−1)^{j₁+j₂+j₃}. - AZ classification: 2 complex (A, AIII) + 8 real (AI, BDI, D, DIII, AII, CII, C, CI) classes ↔ symmetric spac Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -