论文标题
关于Lie代数$ \ MATHFRAK {G} _2 \ subset \ Mathfrak {so}（7）$，Associative $ 3 $ -PLANES和$ \ MATHFRAK {SO}（4）$ subalgebras的观察结果
Observations about the Lie algebra $\mathfrak{g}_2 \subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras
论文作者
论文摘要
我们进行了几个有关Lie代数$ \ MATHFRAK {G} _2 \ subset \ Mathfrak {so}（7）$，联想$ 3 $ -PLANES和$ \ MATHFRAK {SO}（SO}（4）$ subergebras。有些可能是众所周知的，但在文献中不容易找到，而其他结果是新的。我们表明，一个元素$ x \ in \ mathfrak {g} _2 $不能具有排名$ 2 $，如果其排名$ 4 $，则其内核是一个关联子空间。我们证明了$ \ mathfrak {g} _2 $元素的规范定理。给定一个$ \ mathbb r^7 $中的关联$ 3 $ -PLANE $ P $，我们构建了lie subalgebra $θ（p）$ of $ \ mathfrak {so}（7）=λ^2（\ mathbb r^7）$，这是等于ymomorphic to Isomorphic to Isomorphic to $ \ \ Mathfrak {so so}（so}（4）$。这个$ \ mathfrak {so}（4）$ subalgebra与$ \ mathfrak {so}（4）$ subalgebras的其他已知构造不同，由$ \ mathfrak {so}（so}（7）$确定的关联$ 3 $ - plane。这些是NSERC本科研究项目的结果。该论文是写的，以便可以吸引广泛的观众。
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 $Θ(P)$ of $\mathfrak{so}(7) = Λ^2 (\mathbb R^7)$ 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.
