The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors
Abstract
The conformal geometric algebra Cl(4,2) ~ R(8) carries a natural Z_2-grading defined by the dilatation bivector D=\e_4e_0, which satisfies D2=1 and splits the eight-dimensional spinor module into two four-dimensional eigenspaces S=S+⊕ S-. We prove that the ad_D-eigenvectors of Cl(4,2), those elements with eigenvalue~±1 under the Lie bracket [D,·], are exhausted by 16 doublets of the form T±J=n• eJ, where n• ∈ n∞,no is a null vector and eJ is a product of Lorentz-sector basis vectors indexed by J⊆{1,2,3,5}. These doublets satisfy universal algebraic identities: (T±J)2=0 (nilpotency) and T+J T-J=(−1)k(k+1)/2+1 ηJ 2Π+ (single-projector proportionality), where k=|J|, ηJ=∏j∈ Jηjj is the spectator metric, and Π+=1/2(1+D). The contracted sums ∑|J|=kT+J T-J are shown to be controlled by the elementary symmetric polynomials Sk of the Lorentz signature values (+1,+1,+1,−1), with the vanishing of S2=0 reflecting the (3,1) signature of physical spacetime. The even-grade doublets (belonging to Cl^+(4,2) ~ Cl(4,1)) contribute 8Π+ to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra sl(8,R) in which the conformal algebra so(4,2) embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension Cl(p+1,q+1); there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars ... [shortened]
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BibTeX
@misc{hansen2026doublet,
title = {{The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors}},
author = {Steen H. Hansen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22275},
eprint = {2609.22275},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22275},
}
RIS
TY - UNPB
TI - The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors
AU - Steen H. Hansen
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.22275
UR - https://arxiv.org/abs/2609.22275
AB - The conformal geometric algebra Cl(4,2) ~ R(8) carries a natural Z_2-grading defined by the dilatation bivector D=\e_4e_0, which satisfies D^2=1 and splits the eight-dimensional spinor module into two four-dimensional eigenspaces S=S_+⊕ S_-. We prove that the ad_D-eigenvectors of Cl(4,2), those elements with eigenvalue~±1 under the Lie bracket [D,·], are exhausted by 16 doublets of the form T^±_J=n_• e_J, where n_• ∈ n_∞,n_o is a null vector and e_J is a product of Lorentz-sector basis vectors indexed by J⊆{1,2,3,5}. These doublets satisfy universal algebraic identities: (T^±_J)^2=0 (nilpotency) and T^+_J T^-_J=(−1)^k(k+1)/2+1 η_J 2Π_+ (single-projector proportionality), where k=|J|, η_J=∏_j∈ Jη_jj is the spectator metric, and Π_+=1/2(1+D). The contracted sums ∑_|J|=kT^+_J T^-_J are shown to be controlled by the elementary symmetric polynomials S_k of the Lorentz signature values (+1,+1,+1,−1), with the vanishing of S_2=0 reflecting the (3,1) signature of physical spacetime. The even-grade doublets (belonging to Cl^+(4,2) ~ Cl(4,1)) contribute 8Π_+ to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra sl(8,R) in which the conformal algebra so(4,2) embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension Cl(p+1,q+1); there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars ... [shortened]
ER -