Double derivations of n-Lie algebras

After introducing double derivations of n-Lie algebra L weý ýdescribe the relationship between the algebra D(L) of double derivations and the usualý ýderivation Lie algebra Der(L)ý. ýIn particularý, ýwe prove that the inner derivation algebra ad(L)ý ýis an ideal of the double derivation algebra D(L); we also show that if L is a perfect n-Lie algebraý ýwith certain constraints on the base field then the centralizer of ad(L) in D(L) isý ýtrivial and D(L) is centerlessý. ýIn additioný, ýwe obtain that for every perfect n-Lieý ýalgebra L with zero centerý, ýthe triple derivations of the derivation algebra Der(L) are exactlyý ýthe derivations of Der(L)ý, ýand the triple derivations of the inner derivation algebra ad(L) areý ýprecisely the derivations of ad(L).