Triality over Schemes

TL;DR

This paper develops an alternative approach to triality over schemes using Clifford algebras and group computations, establishing equivalences between stacks of trialitarian structures and torsors of certain algebraic groups.

Contribution

It introduces a new scheme-theoretic framework for triality that avoids octonions, using Clifford algebras and group actions, and establishes equivalences with gerbes of torsors for groups of type D4.

Findings

Defined the stack of trialitarian triples and proved its equivalence to the gerbe of PGO8+--torsors.

Established the group of endomorphisms as isomorphic to S3.

Constructed functors relating trialitarian algebras to Spin and PGO+ groups.

Abstract

Working over an arbitrary base scheme, we provide an alternative development of triality which does not use Octonion algebras or symmetric composition algebras. Instead, we use the Clifford algebra of the split hyperbolic quadratic form of rank 8 and computations with Chevalley generators of groups of type $D_4$. Following the strategy of The Book of Involutions [KMRT], we then define the stack of trialitarian triples and show it is equivalent to the gerbe of $\mathbf{PGO}_8^+$--torsors. We show it has endomorphisms generating a group isomorphic to $\mathbb{S}_3$ and that several familiar cohomological properties of $\mathbf{PGO}_8^+$ follow in this setting as a result. Next, we define the stack of trialitarian algebras and show it is equivalent to the gerbe of $\mathbf{PGO}_8^+\rtimes \mathbb{S}_3$--torsors. Because of this, it is also equivalent to the gerbes of simply connected, respectively adjoint, groups of type $D_4$. We define $\mathbf{Spin}_\mathcal{T}$ and $\mathbf{PGO}^+_\mathcal{T}$ for a trialitarian algebra and define concrete functors $\mathcal{T} \mapsto \mathbf{Spin}_\mathcal{T}$ and $\mathcal{T} \mapsto \mathbf{PGO}^+_\mathcal{T}$ which realize these equivalences.