Generalized derivations of Lie algebras

Suppose L is a finite-dimensional Lie algebra with multiplication mu: L boolean AND L --> L. Let Delta(L) denote the set of triples (f,f',f"), with f,f',f" is an element of Hom(L,L), such that mu o (f boolean AND I(L) + I(L) boolean AND f') = f" o mu. We consider the Lie algebra GenDer(L)= {f is an element of Hom(L, L) \ There Exists f', f": (f, f', f") epsilon Delta (L)}. Well-researched sub algebras of GenDer(L) include the derivation algebra, Der(L) = {f is an element of Hom(L, L) \ (f, f, f) is an element of Delta(L)}, and the centroid, C(L) = {f is an element of Hom(L, L) \ (f, 0, f) is an element of Delta(L)}. We now study the subalgebra QDer(L)= {f is an element of Hom(L, L) \ There Exists f': (f, f, f) is an element of Delta(L)}, and the subspace QC(L)= {f is an element of Hom(L, L) \ (f,-f,0) is an element of Delta(L)}. In characteristic not equal 2, GenDer(L) = QDer(L)+ QC(L) and we are concerned with the inclusions Der(L) subset of or equal to QDer(L) and C(L) subset of or equal to QC(L) boolean AND QDer(L). If Z(L) = 0 then C(L) = QC(L)n 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)+ (I(L)). 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 is an element of Hom(L, L) such that f o mu or mu o (f boolean AND I(L)) defines a Lie multiplication. (C) 2000 Academic Press.