Three-Dimensional Geometry in Exceptional Algebra

Abstract

We review some topics in "exceptional mathematics'' from the perspective of 3-dimensional geometry: the octonions $\mathbb{O}$, the split octonions ${\mathbb{O}}^{\prime }$, the bioctonions ${\mathbb{O}}_{\mathbb{C}}\cong \mathbb{C}{\otimes }_{\mathbb{R}}\mathbb{O}$, the complex Albert algebra ${\mathfrak{h}}_3({\mathbb{O}}_{\mathbb{C}})$, and the complex form of the exceptional Lie algebra ${\mathfrak{e}}_6$. We show how to functorially build an algebra isomorphic to $\mathbb{O}$ from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to ${\mathbb{O}}^{\prime }$ starting from a 3d real vector space equipped with a volume form, and one isomorphic to ${\mathbb{O}}_{\mathbb{C}}$ starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to ${\mathfrak{h}}_3({\mathbb{O}}_{\mathbb{C}})$ starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra ${\mathfrak{e}}_6$ and its subalgebra $\mathfrak{sl}(3,\mathbb{C})\oplus \mathfrak{sl}(3,\mathbb{C})\oplus \mathfrak{sl}(3,\mathbb{C})$.

28 pages LaTeX, TikZ figures