Generalized derivations of Lie algebras (Q1570873)

Let \(L\) be a finite dimensional Lie algebra with multiplication given by \(\mu\). The authors study the generalized derivations of \(L\), \(\text{GenDer}(L)\), consisting of the elements \(f\in\Hom(L,L)\) such that there exist \(f',f''\in\Hom(L,L)\) satisfying \(\mu(f(x),y)+\mu(x,f'(y))=f''(\mu(x,y))\) for all \(x,y\in L\). This set is a Lie algebra and contains the algebra of derivations, \(\text{Der}(L)\), and the centroid, \(C(L)\). The authors also study the quasiderivations of \(L\), \(Q\operatorname {Der}(L)\), consisting of the elements \(f\in \Hom(L,L)\) such that there exists \(f'\in \Hom(L,L)\) satisfying \(\mu(f(x),y)+\mu(x,f(y))=f'(\mu(x,y))\) for all \(x,y\in L\) and the quasicentroid of \(L\), \(QC(L)\), consisting of the elements \(f\in \Hom(L,L)\) satisfying \(\mu(f(x),y)=\mu(x,f(y))\) for all \(x,y\in L\). The authors prove a number of theorems regarding relations among these objects. For instance, in characteristic not equal to \(2\), GenDer\((L)=Q\text{Der}(L)+QC(L)\). If it is also assumed that the center of \(L\) is trivial, then \(C(L)=QC(L)\cap Q\operatorname {Der}(L)\) and, under reasonable conditions on Lie algebras with toral Cartan subalgebras, they also show \(Q\operatorname {Der}(L)=\operatorname {Der}(L)+C(L)\). In the special case of characteristic \(0\) and \(L\) a parabolic subalgebra of a simple Lie algebra of rank bigger than \(1\), the authors prove \(\operatorname {GenDer}(L)=ad(L)+(I_L)\). Again assuming that the center of \(L\) is trivial, it is shown that \(QC(L)\) is a commutative, associative algebra and conditions are given forcing \(QC(L)=C(L)\) and \(\operatorname {GenDer}(L)=Q\operatorname {Der}(L)\). Additionally, in characteristic \(0\), they show \(\operatorname {GenDer}(L)\) preserves the radical of \(L\). Finally they apply some of these results to study functions \(f\in \Hom(L,L)\) satisfying \(f\circ\mu\) or \(\mu\circ(f\wedge I_L)\) defines a Lie multiplication.