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. EXPONENTIALS AND SANDWICH ELEMENTSLet g be a finite-dimensional Lie algebra over k. An element c ∈ g is called an absolute zerodivisor if (ad c) 2 = 0 (see [39]). In more recent terminology, c is sometimes referred to as a sandwich Lemma 1.1.1. The following statements hold:(1) If x ∈ N p+1 2 (g), then exp(x) ∈ Aut(g).(2) If x, y ∈ N p+1 2 (g) are such that [x, y] = 0, then exp(x+y) = exp(x) • exp(y).Note that the subsets Sw(g) ⊆ N p+1 2 (g) ⊆ N p (g) are stable with respect to the canonical action of Aut(g) on g. Moreover, we haveWe denote by C(g) the center of g and call g centerless in case C(g) = (0). We say that S ⊆ g is a Lie subset, provided [s, t] ∈ S for all s, t ∈ S.The automorphism group of a restricted Lie algebra (g, [p]) will be denoted Aut p (g). As it coincides with the stabilizer of the p-map, Aut p (g) is a closed subgroup of Aut(g) ⊆ GL(g). We let G g := Aut p (g) • be the identity component of Aut p (g).If (g, [p]) is centerless, then N p (g) = V (g) is the nullcone of g.Lemma 1.1.2. Let g be a Lie algebra.(1) If X ⊆ g is a conical closed Aut(g)-stable subset, then [c, x] ∈ X for all c ∈ Sw(g) and x ∈ X.(2) Sw(g) is a Lie subset of g.(3) If (g, [p]) is restricted and centerless, then Sw(g) ⊆ V (g) is a Lie subset and exp(Sw(g)) ⊆ G g .(4) If (g, [p]) is restricted and n ✂ g is an elementary abelian p-ideal, then exp(n) ⊆ G g is an abelian, connected subgroup.