Octonionic representations of Clifford algebras and triality
Jörg Schray, Corinne A. Manogue
TL;DR
This work develops octonionic representations of Clifford algebras, introducing octonionic spinors and their bilinear covariants to realize orthogonal groups; it shows that non-associativity can be managed via the Clifford group and the use of opposite algebras, notably in constructing a fully octonionic description of triality. Central to the approach is the octonionic representation of ${\cal Cl}(8,0)$ and its reduction/extension to related algebras ${\cal Cl}_0(8,0)$, ${\cal Cl}(0,7)$, and ${\cal Cl}(9,1)$, all expressed through octonionic matrices and minimal left ideals. The Chevalley algebra $\mathcal A = V \oplus S_0 \oplus S_1$ is recast in an octonionic framework via 3×3 Hermitian octonionic matrices, making triality an intrinsic symmetry: the triality maps generate a $\Sigma_3$ action commuting with $SO(8)$, i.e., $\Sigma_3 \times SO(8)$. The paper also argues for a finite-generator perspective on group structure in the octonionic setting, outlining practical constructions for both vector and spinor representations in higher dimensions, with implications for supersymmetric theories. Overall, the octonionic formalism yields a coherent, highly symmetric view of Clifford-algebra representations, spinors, and triality that could clarify the symmetries of advanced theoretical frameworks like superstrings and supergravity.
Abstract
The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework.