Triality over schemes
The article under review provides an alternative development of triality that does not use octonion algebras or symmetric composition algebras. Instead, the Clifford algebra of the split hyperbolic quadratic form of rank 8 and computations with Chevalley generators of groups of type \( D_{4} \) are used. The setting and notational system whithin which the article has been written can be found in [\textit{P. Gille} et al., ``Azumaya algebras and obstructions to quadratic pairs over a scheme'', Preprint, \url{arxiv:2209.07107}; \textit{C.~Ruether}, ``The canonical quadratic pair on Clifford algebras over schemes'', Preprint, \url{arXiv:2309.03077}]. In the paper, general sheaves are denoted by calligraphic letters, bold letters mean group sheaves, other rings and algebras are denoted by roman letters.\N\NThe preliminary part of the paper gives the setting over a scheme as well the definitions and background results for main objects studied in the paper such as quadratic triples, the groups \( \mathbf{Spin} ( \mathcal{A}, \sigma, f), \mathbf{O}^{+} ( \mathcal{A}, \sigma, f) \) and \( \mathbf{PGO}^{+} ( \mathcal{A}, \sigma, f) \), where \(( \mathcal{A}, \sigma, f) \) denotes an Azumaya algebra with quadratic pair over a field. The stack of trialitarian triples is defined and it is shown that it is equivalent to the gerbe of \( \mathbf{PGO}_{8}^{+} \)-torsors. It is shown also that it has endomorphisms generating a group isomorphic to the symmetric group \( \mathbb{S}_{3} \) and that several familiar cohomological properties of \( \mathbf{PGO}_{8}^{+} \) follow in this setting. Further, the author defines the stack of trialitarian algebras and shows that it is equivalent to the gerbe of \( \mathbf{PGO}_{8}^{+}\rtimes \mathbb{S}_{3} \)-torsors. For this reason, it is also equivalent to the gerbes of simply connected, respectively adjoint, groups of type \( D_{4} \). For a trialitarian algebra \( \mathcal{T}\), the author defines \( \mathbf{Spin}_{\mathcal{T}} \) and \( \mathbf{PGO}^{+}_{\mathcal{T}} \) establishing then concrete functors \( \mathcal{T}\to\mathbf{Spin}_{\mathcal{T}} \) and \( \mathcal{T}\to\mathbf{PGO}^{+}_{\mathcal{T}} \) that realize these equivalences.
scientific article; zbMATH DE number 7958115
scientific article