English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

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 otherresults are new. We show that an element $X \in \mathfrak{g}_2$ cannot haverank $2$, and if it has rank $4$ then its kernel is an associative subspace. Weprove a canonical form theorem for elements of $\mathfrak{g}_2$. Given anassociative $3$-plane $P$ in $\mathbb R^7$, we construct a Lie subalgebra$\Theta(P)$ of $\mathfrak{so}(7) = \Lambda^2 (\mathbb R^7)$ that is isomorphicto $\mathfrak{so}(4)$. This $\mathfrak{so}(4)$ subalgebra differs from otherknown constructions of $\mathfrak{so}(4)$ subalgebras of $\mathfrak{so}(7)$determined by an associative $3$-plane. These are results of an NSERCundergraduate research project. The paper is written so as to be accessible toa wide audience.

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