Double derivations of n-Lie algebras

After introducing double derivations of n-Lie algebra L we describe the relationship between the algebra D(L) of double derivations and the usual derivation Lie algebra Der(L). In particular, we prove that the inner derivation algebra ad(L) is an ideal of the double derivation algebra D(L); we also show that if L is a perfect n-Lie algebra with cer- tain constraints on the base eld then the centralizer of ad(L) in D(L) is trivial and D(L) is centerless. In addition, we obtain that for every perfect n-Lie algebra L with zero center, the triple derivations of the derivation algebra Der(L) are exactly the derivations of Der(L), and the triple derivations of the inner derivation algebra ad(L) are precisely the derivations of ad(L).