Moufang sets and structurable division algebras

TL;DR

This paper establishes a deep connection between structurable division algebras and Moufang sets, providing explicit formulas and extending known results from Jordan algebras to a broader class.

Contribution

It proves that every structurable division algebra induces a Moufang set and characterizes Moufang sets from simple algebraic groups as arising from structurable division algebras.

Findings

Every structurable division algebra gives rise to a Moufang set.

Moufang sets from simple algebraic groups of rank one correspond to structurable division algebras.

Explicit formulas for root groups, -map, and Hua maps are provided.

Abstract

A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. We also obtain explicit formulas for the root groups, the \tau-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.