Varieties of elementary abelian Lie algebras and degrees of modules

Abstract

Let(g,[p])(\mathfrak {g},[p])be a restricted Lie algebra over an algebraically closed fieldkkof characteristicp≥3p\!\ge \!3. Motivated by the behavior of geometric invariants of the so-called(g,[p])(\mathfrak {g},[p])-modules of constantjj-rank (j∈{1,…,p−1}j \in \{1,\ldots ,p\!-\!1\}), we study the projective varietyE(2,g)\mathbb {E}(2,\mathfrak {g})of two-dimensional elementary abelian subalgebras. Ifp≥5p\!\ge \!5, then the topological spaceE(2,g/C(g))\mathbb {E}(2,\mathfrak {g}/C(\mathfrak {g})), associated to the factor algebra ofg\mathfrak {g}by its centerC(g)C(\mathfrak {g}), is shown to be connected. We give applications concerning categories of(g,[p])(\mathfrak {g},[p])-modules of constantjj-rank and certain invariants, calledjj-degrees.