Varieties of Elementary Abelian Lie Algebras and Degrees of Modules

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.