Trialitarian Triples

TL;DR

This paper explicitly describes the canonical isomorphisms in trialitarian triples of degree 8 algebras with involution, using a rational shift operator that cyclically permutes the algebras, applicable over any field characteristic.

Contribution

It introduces a rationally defined shift operator to explicitly construct trialitarian isomorphisms, unifying the structure of trialitarian triples without characteristic restrictions.

Findings

Explicit description of trialitarian isomorphisms using a shift operator

Construction relies on compositions of 8-dimensional quadratic spaces

Applicable to all base field characteristics

Abstract

Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical "trialitarian'' isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.