Involutive Duality of Lie Operators in the Extended Alpha Group
Abstract
This work investigates the involutive, adjoint, and spectral structures arising in the extended Alpha Group. Starting from the four-dimensional Alpha Group transformation matrix, whose angular depen- dence is governed by coupled tan θ and cot θ modes, an iterative commutator procedure generates a sixteen-dimensional real Lie algebra. Within this algebra, the operators B5 and B6 are identified as structurally distinguished elements and analyzed through their adjoint actions. The operator B6 is veri- fied to be involutive and induces an 8 + 8 decomposition of the Lie algebra, while the adjoint action of B5 defines an eight-dimensional invariant mixed sector through a polynomial kernel. The restricted adjoint operators preserve this sector, and their coupled action generates an idempotent operator Q, satisfying Q2 = Q, revealing an invariant projective structure. The spectral analysis further identifies characteristic- polynomial factorizations associated with B5 , B6 , and their compositions, including golden-ratio relations among distinct generators of the extended algebra. These results establish structural connections between the original modal representation, Lie-algebraic involution, adjoint dynamics, projective organization, and the spectral structure of the extended Alpha Group.
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BibTeX
@misc{correa2026involutive,
title = {{Involutive Duality of Lie Operators in the Extended Alpha Group}},
author = {Cleber Souza Correa and Thiago Braido Nogueira de Melo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.14547},
eprint = {2611.14547},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.14547},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Involutive Duality of Lie Operators in the Extended Alpha Group
AU - Cleber Souza Correa
AU - Thiago Braido Nogueira de Melo
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.14547
UR - https://arxiv.org/abs/2611.14547
AB - This work investigates the involutive, adjoint, and spectral structures arising in the extended Alpha Group. Starting from the four-dimensional Alpha Group transformation matrix, whose angular depen- dence is governed by coupled tan θ and cot θ modes, an iterative commutator procedure generates a sixteen-dimensional real Lie algebra. Within this algebra, the operators B5 and B6 are identified as structurally distinguished elements and analyzed through their adjoint actions. The operator B6 is veri- fied to be involutive and induces an 8 + 8 decomposition of the Lie algebra, while the adjoint action of B5 defines an eight-dimensional invariant mixed sector through a polynomial kernel. The restricted adjoint operators preserve this sector, and their coupled action generates an idempotent operator Q, satisfying Q2 = Q, revealing an invariant projective structure. The spectral analysis further identifies characteristic- polynomial factorizations associated with B5 , B6 , and their compositions, including golden-ratio relations among distinct generators of the extended algebra. These results establish structural connections between the original modal representation, Lie-algebraic involution, adjoint dynamics, projective organization, and the spectral structure of the extended Alpha Group.
ER -