A₁ as the Universal Lie-Algebraic Skeleton of Qubit Symmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The A₁ root system (sl(2,C)) is the fundamental Lie-algebraic skeleton of qubit Hilbert space, with its principal elements and embeddings into sl(m,K) revealing universal structural constraints; Malcev algebras and Nichols braided Lie algebras extend this to non-associative and braided quantum symmetries. | MATH: A₁ ≅ sl(2,C) has root system {±α}, Cartan matrix [2], Weyl group ℤ₂; spin-1/2 Hilbert space ℂ² is the fundamental irrep with Pauli generators σ_x, σ_y, σ_z satisfying [σ_i,σ_j]=2iε_ijk σ_k. Frobenius Lie algebra principal element: e = Σ h_i (Cartan subalgebra generator) with semisimple part having eigenvalues related to root values; embedding theorem: any Lie algebra with left-symmetric (pre-Lie) structure embeds into sl(m,K) — for A₁, m=2 gives the qubit. Malcev algebra correspondence: a Lie algebra L(M) uniquely determined by Malcev algebra M, with anti-commutator [x,y] and Jacobiator proportional to associator — for A₁, the Jacobi identity holds trivially. Nichols
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 as the Universal Lie-Algebraic Skeleton of Qubit Symmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587193},
url = {https://zenodo.org/records/23587193},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A₁ as the Universal Lie-Algebraic Skeleton of Qubit Symmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587193
UR - https://zenodo.org/records/23587193
AB - FINDING: The A₁ root system (sl(2,C)) is the fundamental Lie-algebraic skeleton of qubit Hilbert space, with its principal elements and embeddings into sl(m,K) revealing universal structural constraints; Malcev algebras and Nichols braided Lie algebras extend this to non-associative and braided quantum symmetries. | MATH: A₁ ≅ sl(2,C) has root system {±α}, Cartan matrix [2], Weyl group ℤ₂; spin-1/2 Hilbert space ℂ² is the fundamental irrep with Pauli generators σ_x, σ_y, σ_z satisfying [σ_i,σ_j]=2iε_ijk σ_k. Frobenius Lie algebra principal element: e = Σ h_i (Cartan subalgebra generator) with semisimple part having eigenvalues related to root values; embedding theorem: any Lie algebra with left-symmetric (pre-Lie) structure embeds into sl(m,K) — for A₁, m=2 gives the qubit. Malcev algebra correspondence: a Lie algebra L(M) uniquely determined by Malcev algebra M, with anti-commutator [x,y] and Jacobiator proportional to associator — for A₁, the Jacobi identity holds trivially. Nichols Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -