Varieties of elementary abelian Lie algebras and degrees of modules

Abstract

Let (g, [p]) be a restricted Lie algebra over an algebraically closed field k of characteristic p ≥ 3. Motivated by the behavior of geometric invariants of the so-called (g, [p])-modules of constant j-rank (j ∈ {1,. .. , p − 1}), we study the projective variety E(2, g) of two-dimensional elementary abelian subalgebras. If p ≥ 5, then the topological space E(2, g/C(g)), associated to the factor algebra of g by its center C(g), is shown to be connected. We give applications concerning categories of (g, [p])-modules of constant j-rank and certain invariants, called j-degrees.