An Algebraic Description of the Exceptional Isogenies to Orthogonal Groups

TL;DR

This paper provides an algebraic framework for understanding exceptional isogenies between classical groups and orthogonal groups of small dimensions, utilizing explicit constructions based on the Cartan--Dieudonné Theorem.

Contribution

It introduces a unified algebraic approach to describe these isogenies, including new ones related to triality, through quaternion and bi-quaternion algebra constructions.

Findings

Explicit algebraic constructions of exceptional isogenies

Unified approach encompassing quaternion and bi-quaternion algebras

Identification of new isogenies related to triality

Abstract

We show how the exceptional isogenies of classical groups to orthogonal groups of quadratic spaces of dimensions up to 8 over fields of characteristic different from 2 may be obtained by explicit algebraic constructions using the Cartan--Dieudonn\'{e} Theorem. This gives a unified approach to groups arising from quaternion algebras, bi-quaternion algebras, and matrices over bi-quaternion algebras. The latter seem to yield new isogenies, which are related to triality in the hyperbolic case.