The Doublet Tower of $$\text {Cl}(4,2)$$: Universal Nilpotency, Projector Identities, and the $$\mathfrak {sl}(8,\mathbb {R})$$ Parabolic from Conformal Null Vectors
Year: 2026
Type: article
Abstract: The conformal geometric algebra $$\text {Cl}(4,2)\cong \mathbb {R}(8)$$ Cl ( 4 , 2 ) ≅ R ( 8 ) carries a natural $$\mathbb {Z}_2$$ Z 2 -grading defined by the dilatation bivector $$D=\varvec{e}_{4}\varvec{e}_{0}$$ D = e 4 e 0 , which satisfies $$D^2=1$$ D 2 = 1 and splits the eight-dimensional spinor module into two four-dimensional eigenspaces $$S=S_+\oplus S_-$$ S = S + ⊕ S - . We prove that the $$\text {ad}_D$$ ad D -eigenvectors of $$\text {Cl}(4,2)$$ Cl ( 4 , 2 ) , those elements with eigenvalue $$\pm 1$$ ± 1 under the Lie bracket $$[D,\cdot ]$$ [ D , · ] , are exhausted by 16 doublets of the form $$T^\pm _J=n_\bullet e_J$$ T J ± = n ∙ e J , where $$n_\bullet \in \{n_{\infty },n_{\text {o}}\}$$ n ∙ ∈ { n ∞ , n o } is a null vector and $$e_J$$
Author: Steen H. Hansen
Institution: University of Copenhagen
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7223089089