Varieties of Elementary Abelian Lie Algebras and Degrees of Modules

TL;DR

This paper investigates the structure of elementary abelian subalgebras in restricted Lie algebras over algebraically closed fields, revealing connectivity properties and applications to module invariants like $j$-degrees.

Contribution

It introduces the variety of two-dimensional elementary abelian subalgebras and proves its connectedness for certain Lie algebras, with applications to module invariants.

Findings

The variety $ ext{E}(2, extbf{g}/C( extbf{g}))$ is connected for $p \\ge 5$.

Applications to categories of modules with constant $j$-rank.

Insights into $j$-degrees of modules.

Abstract

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.