Moufang sets arising from polarities of Moufang planes over octonion division algebras

TL;DR

This paper classifies all polarities with absolute points in octonion planes, revealing four types of Moufang sets linked to algebraic groups of types F4, 2E6, C4, and special projective sets, with distinctions based on field characteristics.

Contribution

It provides a complete classification of polarities and associated Moufang sets in octonion planes, including explicit descriptions of new Moufang sets of type 2E6.

Findings

Four types of polarities lead to distinct Moufang sets.

Moufang sets of type 2E6 correspond to algebraic groups of type 2E6,1^29.

Characteristic of the field determines the occurrence of certain Moufang sets.

Abstract

For every octonion division algebra O, there exists a projective plane which is parametrized by O; these planes are related to rank two forms of linear algebraic groups of absolute type E6. We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set. It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F4, (2) Moufang sets of type 2E6, (3) hermitian Moufang sets of type C4, and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra. Case (3) only occurs over fields of characteristic different from two, whereas case (4) only occurs over fields of characteristic equal to two. The Moufang sets of type 2E6 that we obtain in case (2) are exactly those corresponding to linear algebraic groups of type 2E6,1^29; the explicit description of those Moufang sets was not yet known.