Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

Abstract

From an octonion algebra $\mathbb{O}$ over a field $k$ of characteristic not two or three, we show that the fundamental representation ${\rm Im}(\mathbb{O})$ of the derivation algebra ${\rm Der}(\mathbb{O})$ and the spinor representation $\mathbb{O}$ of $\mathfrak{so}({\rm Im}(\mathbb{O}))$ are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics and we show that the covariants of these representations and their Mathews identities are related to the Fano plane and the affine space $(\mathbb{Z}_2)^3$. This also permits to give constructions of exceptional Lie superalgebras.