Observations about the Lie algebra ${\mathfrak{g}}_2\subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras

Abstract

We make several observations relating the Lie algebra ${\mathfrak{g}}_2\subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in {\mathfrak{g}}_2$ cannot have rank $2$, and if it has rank $4$ then its kernel is an associative subspace. We prove a canonical form theorem for elements of ${\mathfrak{g}}_2$. Given an associative $3$-plane $P$ in ${\mathbb{R}}^7$, we construct a Lie subalgebra $\Theta (P)$ of $\mathfrak{so}(7)={\Lambda }^2({\mathbb{R}}^7)$ that is isomorphic to $\mathfrak{so}(4)$. This $\mathfrak{so}(4)$ subalgebra differs from other known constructions of $\mathfrak{so}(4)$ subalgebras of $\mathfrak{so}(7)$ determined by an associative $3$-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.

21 pages, no figures. These are the results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience. Comments are welcome. Version 2: minor typos corrected as per referee's report. Final version, to appear in "Expositiones Mathematicae"

References in corpus (1)

• Flows of G_2 Structures, I