Generalized Derivations of Lie Algebras

Suppose L is a finite-dimensional Lie algebra with multiplication μ: L L → L. Let Δ(L) denote the set of triples (f, f′, f″), with f, f′, f″ Hom(L, L), such that μ (f IL + IL f′) = f″ μ. We consider the Lie algebra GenDer(L) = {f Hom(L, L) f′, f″: (f, f′, f″) Δ(L)}. Well-researched subalgebras of GenDer(L) include the derivation algebra, Der(L) = {f Hom(L, L)(f, f, f) Δ(L)}, and the centroid, C(L) = {f Hom(L, L)(f, 0, f) δ(L)}. We now study the subalgebra QDer(L) = {f Hom(L, L) f′: (f, f, f′) Δ(L)}, and the subspace QC(L) = {f Hom(L, L)(f, −f, 0) Δ(L)}. In characteristic ≠ 2, GenDer(L) = QDer(L) + QC(L) and we are concerned with the inclusions Der(L) QDer(L) and C(L) QC(L) ∩ QDer(L). If Z(L) = 0 then C(L) = QC(L) ∩ QDer(L) and, under reasonable conditions on Lie algebras with toral Cartan subalgebras, we show QDer(L) = Der(L) + C(L); if L is a parabolic subalgebra of a simple Lie algebra of rank > 1 in characteristic 0, then we even have GenDer(L) = ad(L) + (IL). In general QC(L) is not closed under composition or Lie bracket; however, if Z(L) = 0 then QC(L) is a commutative, associative algebra, and we describe conditions that force QC(L) = C(L) or, equivalently, GenDer(L) = QDer(L). We show that, in characteristic 0, GenDer(L) preserves the radical of L, thus generalizing the classical result for Der(L). We also discuss some applications of the main results to the study of functions f Hom(L, L) such that f μ or μ (f IL) defines a Lie multiplication