Malcev Algebras' Largest Lie Subalgebra and Qubit Hilbert Space Structure — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Malcev algebras admit a canonical largest Lie subalgebra, enabling a unique Lie-algebra correspondence that may underpin qubit Hilbert-space structure via associator/Jacobiator deviations. | MATH: Malcev identity: [[x,y],[z,w]] = [[[x,y],z],w] + [[[y,z],x],w] + [[[z,x],y],w] + [[[x,w],z],y] + [[[y,w],x],z] + [[[z,w],x],y] (generalized Jacobi). Lie subalgebra L(M) = elements whose left multiplication is a derivation; Jacobiator vanishes on L(M). Qubit Hilbert space ℂ2 with Pauli basis {iσx,iσy,iσz} forms su(2)≅so(3), a Lie algebra — not Malcev (Jacobi holds). Associator in non-associative algebras: A(x,y,z)=(xy)z−x(yz); for Malcev, associator is alternating and related to Jacobiator J(x,y,z)=[[x,y],z]+[[y,z],x]+[[z,x],y]. | CONNECTION: The Lie algebra su(2) has root system A1 — a single root pair ±α with Cartan matrix [2]. Weyl group ℤ2. The r
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026malcev,
title = {{Malcev Algebras' Largest Lie Subalgebra and Qubit Hilbert Space Structure — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635295},
url = {https://zenodo.org/records/23635295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Malcev Algebras' Largest Lie Subalgebra and Qubit Hilbert Space Structure — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635295
UR - https://zenodo.org/records/23635295
AB - FINDING: Malcev algebras admit a canonical largest Lie subalgebra, enabling a unique Lie-algebra correspondence that may underpin qubit Hilbert-space structure via associator/Jacobiator deviations. | MATH: Malcev identity: [[x,y],[z,w]] = [[[x,y],z],w] + [[[y,z],x],w] + [[[z,x],y],w] + [[[x,w],z],y] + [[[y,w],x],z] + [[[z,w],x],y] (generalized Jacobi). Lie subalgebra L(M) = elements whose left multiplication is a derivation; Jacobiator vanishes on L(M). Qubit Hilbert space ℂ^2 with Pauli basis {iσ_x,iσ_y,iσ_z} forms su(2)≅so(3), a Lie algebra — not Malcev (Jacobi holds). Associator in non-associative algebras: A(x,y,z)=(xy)z−x(yz); for Malcev, associator is alternating and related to Jacobiator J(x,y,z)=[[x,y],z]+[[y,z],x]+[[z,x],y]. | CONNECTION: The Lie algebra su(2) has root system A_1 — a single root pair ±α with Cartan matrix [2]. Weyl group ℤ_2. The r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -