A new construction of Moufang quadrangles of type E6, E7 and E8

TL;DR

This paper introduces a novel construction method for Moufang quadrangles of types E6, E7, and E8 using tensor products of composition algebras, providing a more systematic approach than previous ad-hoc methods.

Contribution

It presents a new construction technique for exceptional Moufang quadrangles via tensor products of composition algebras, extending to a general method using Jordan algebra modules.

Findings

Constructs Moufang quadrangles of types E6, E7, E8 in characteristic ≠ 2

Provides a systematic construction method using tensor products

Offers a general approach from Jordan algebra modules

Abstract

In the classification of Moufang polygons by J. Tits and R. Weiss, the most intricate case is by far the case of the exceptional Moufang quadrangles of type E_6, E_7 and E_8, and in fact, the construction that they present is ad-hoc and lacking a deeper explanation. We will show how tensor products of two composition algebras can be used to construct these Moufang quadrangles in characteristic different from 2. As a byproduct, we will obtain a method to construct any Moufang quadrangle in characteristic different from two from a module for a Jordan algebra.