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

Abstract

We continue our study and classification of structures on the Fano plane $\mathcal{F}$ and its dual ${\mathcal{F}}^{\ast }$ 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": $\mathcal{F}\times \mathcal{F}\to \{-1,1\}$, inducing an octonion multiplication, and a function ${\delta }^{\ast }:Aut(\mathcal{F})\times {\mathcal{F}}^{\ast }\to \{-1,1\}$ such that $g\in Aut(\mathcal{F})$ can be lifted to an automorphism of the octonions iff ${\delta }^{\ast }(g,\cdot )$ is the Radon transform of a function on $\mathcal{F}$. We lift the action of $Aut(\mathcal{F})$ on $\mathcal{F}$ to the action of a non-trivial eight-fold covering $Aut(\mathcal{F})$ on a twofold covering $\hat{\mathcal{F}}$ of $\mathcal{F}$ 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 $\mathcal{F}$ and $\epsilon$.