初等交换李代数簇与模的度
Varieties of Elementary Abelian Lie Algebras and Degrees of Modules
摘要：
设$(\mathfrak{g},[p])$是特征为$p\!\ge \!3$的代数闭域$k$上的限制李代数.受常秩$(\mathfrak{g},[p])$ -模（$j \in \{1,\ldots,p\!-\!1\}$）的几何不变量行为的启发，我们研究了二维初等交换子代数的投射簇$\mathbb{E}(2,\mathfrak{g})$.如果$p\!\ge\!5$，则拓扑空间$\mathbb{E}(2,\mathfrak{g}/C(\mathfrak{g}))$，通过其中心$C(\mathfrak{g})$与$\mathfrak{g}$的因子代数相关联，被证明是连通的。我们给出了关于$(\mathfrak{g},[p])$ -模范畴的应用，这些模具有常$j$ -秩和某些不变量，称为$j$ -度。
英文摘要：
Let $(\mathfrak{g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak{g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots,p\!-\!1\}$), we study the projective variety $\mathbb{E}(2,\mathfrak{g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge\!5$, then the topological space $\mathbb{E}(2,\mathfrak{g}/C(\mathfrak{g}))$, associated to the factor algebra of $\mathfrak{g}$ by its center $C(\mathfrak{g})$, is shown to be connected. We give applications concerning categories of $(\mathfrak{g},[p])$-modules of constant $j$-rank and certain invariants, called $j$-degrees.
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