N-derivations Of Lie Algebras

It is well known that derivations are important subjects in Lie algebras. Suppose L a Lie algebra and δ a linear transformation of L, if for any a,b € L,δ([a,b])= [δ(a),b]+[a,δ(b)], then we call δ a derivation of L. Der(L), the set of all derivations of a Lie algebra L, is a Lie algebra under the Lie bracket operation. There arc many reasons for which we study derivations of Lie algebras, for instance, we can describe the structures of Lie algebras and construct Lie algebras by derivations. So, many scholars get interested in derivations strongly and many studies are about generalizations of derivations. There are various generalizations of derivations, such as generalized derivations, triple derivations and n-derivations. So far, there have appeared many interesting results on generalized derivations and triple derivations. Leger and Lucks[4] proved that if the generalized derivations algebras equal its general linear Lie algebras, then L is a two dimension abelian Lie algebra or a three dimension simple Lie algebra. In many cases, all triple derivations of Lie algebras are derivations. If 2 is invertible in the coefficient ring R, then all the triple derivations of a centerless perfect Lie algebra L over R are derivations of L. All the triple derivations of a parabolic subalgebra P of a finite dimension simple Lie algebra L over an algebraically closed field F of characteristic zero are inner derivations of L. Besides, many scholars studied the Lie n-derivations of generalized matrix algebras、triangular rings and triangular algebras.The present article focuses on the study of n-derivations of Lie algebras and Lie supcralgebras. N-derivations are the generalizations of derivations and triple derivations. The same as derivations and triple derivations, we can depict the structures of Lie algebras by n-derivations. At the same time, they have backgrouds of Lie groups, ring theories and operator algebras.The present article describes n-derivations algebras of the following Lie algebras and Lie superalgcbras.1 、centerless perfect Lie algebras; 2、the holomorph of centerless perfect Lie algebras; 3、parabolic subalgebras of finite dimension semisimple Lie algebras; 4、centerless perfect Lie superalgebras. Under appropriate assumption, we prove that all n-dcrivations of such algebras are derivations. At last, we give an example to indicate that all n-derivations of any Lie algebra are not derivations.