Commutation relations of $\mathfrak g\_2$ and the incidence geometry of the Fano plane

TL;DR

This paper explores the algebraic and geometric structures on the Fano plane related to octonions and the Lie algebra g2, revealing new connections between incidence geometry, automorphisms, and algebraic operations.

Contribution

It introduces a novel framework linking the Fano plane's incidence geometry with the structure of octonions and the Lie algebra g2, including automorphism lifts and geometric representations.

Findings

Defined a composition factor inducing octonion multiplication.

Established conditions for automorphism lifts via Radon transform.

Connected incidence geometry with Lie algebra generators.

Abstract

We continue our study and classification of structures on the Fano plane ${\cal F}$ and its dual ${\cal F}^\ast$ involved in the construction of octonions and the Lie algebra $\mathfrak g_2 (\mathbb F)$ over a field $\mathbb F$. These are a "composition factor": ${\cal F}\times {\cal F} \to\{-1, 1\}$, inducing an octonion multiplication, and a function $\delta^\ast : Aut({\cal F}) \times {\cal F}^\ast \to \{-1, 1\}$ such that $g \in Aut({\cal F})$ can be lifted to an automorphism of the octonions iff $\delta^\ast(g, \cdot)$ is the Radon transform of a function on ${\cal F}$. We lift the action of $Aut({\cal F})$ on ${\cal F}$ to the action of a non-trivial eight-fold covering $Aut({\cal F})$ on a twofold covering $\hat {\cal F}$ of ${\cal F}$ contained in the octonions. This extends tautologically to an action on the octonions by automorphism. Finally, we associate to incident point-line pairs a generating set of $\mathfrak g_2 (\mathbb F)$ and express brackets in terms of the incidence geometry of ${\cal F}$ and $\epsilon$.