Inner ideals and structurable algebras: Moufang sets, triangles and hexagons

TL;DR

This paper constructs Moufang sets, triangles, and hexagons from inner ideals of Lie algebras derived from various structurable algebras, linking algebraic structures to geometric objects.

Contribution

It introduces a unified method to generate Moufang geometries from different classes of structurable algebras via the Tits--Kantor--Koecher construction.

Findings

Constructed Moufang sets, triangles, and hexagons from structurable algebras.

Explicitly determined root groups in terms of the underlying algebra.

Connected algebraic structures with geometric configurations.

Abstract

We construct Moufang sets, Moufang triangles and Moufang hexagons using inner ideals of Lie algebras obtained from structurable algebras via the Tits--Kantor--Koecher construction. The three different types of structurable algebras we use are, respectively: (1) structurable division algebras, (2) algebras $D \oplus D$ for some alternative division algebra $D$, equipped with the exchange involution, (3) matrix structurable algebras $M(J,1)$ for some cubic Jordan division algebra $J$. In each case, we also determine the root groups directly in terms of the structurable algebra.