Let $\mathbb{F}$ be a field of characteristic different from two. Suppose that ${\mathcal{N}}_2$ is the category of finite-dimensional nilpotent Lie algebras of class two over the field $\mathbb{F}$ and that $\mathcal{ALT}$ is the category of alternating bilinear maps of $\mathbb{F}$-vector spaces. We establish a relation between the category ${\mathcal{N}}_2$ and the category $\mathcal{ALT}$. Then we show that the problem of determining the capability of these Lie algebras reduces to determining the epicenter of the corresponding objects in $\mathcal{ALT}$. As an application of this technique, we describe the structure of Lie algebras corresponding to alternating bilinear maps of rank one (that is, to alternating bilinear forms). Also, we describe the epicenter of decomposable nondegenerate alternating bilinear maps of rank two.

让 $\mathbb{F}$是一个不同于两个的特征领域。假设${\mathcal{N}}_2$ 是域上第二类有限维幂零李代数的范畴 $\mathbb{F}$ 然后 $\mathcal{ALT}$ 是交替双线性映射的范畴 $\mathbb{F}$-向量空间。我们建立类之间的关系${\mathcal{N}}_2$ 和类别 $\mathcal{ALT}$. 然后我们证明，确定这些李代数的能力的问题简化为确定相应对象的震中$\mathcal{ALT}$. 作为该技术的应用，我们描述了对应于秩为 1 的交替双线性映射（即交替双线性形式）的李代数的结构。此外，我们描述了二阶可分解非退化交替双线性映射的中心。