Double Derivations of n-Lie Superalgebras

Abstract

Let 𝔤 be an n-Lie superalgebra. We study the double derivation algebra [Formula: see text] and describe the relation between [Formula: see text] and the usual derivation Lie superalgebra Der(𝔤). We show that the set [Formula: see text] of all double derivations is a subalgebra of the general linear Lie superalgebra gl(𝔤) and the inner derivation algebra ad(𝔤) is an ideal of [Formula: see text]. We also show that if 𝔤 is a perfect n-Lie superalgebra with certain constraints on the base field, then the centralizer of ad(𝔤) in [Formula: see text] is trivial. Finally, we give that for every perfect n-Lie superalgebra 𝔤, the triple derivations of the derivation algebra Der(𝔤) are exactly the derivations of Der(𝔤).

Double derivations of $n-$Hom-Lie color algebras

“…It is obvious that the derivations of an n−Hom-Lie color algebra L are double derivations, and hence we have Der(L) ⊂ D(L). Note that there exists an n−Lie superalgebra L such that Der(L) = D(L) (see Remark 2.5 in [27]). Proof.…”

“…Note that, if L is not a perfect multiplicative n−Hom-Lie color algebra, then Inn(L) may not be an ideal of D(L) (see Rimark 2.8 in [27]).…”

“…The main purpose of our work is to generalize the results of R. Bai, Y. Gao, Y. Zhang and Z. Li [7] and B. Sun, L. Chen and X. Zhou [27] to the case of multiplicative n−Hom-Lie color algebras. Double derivations are similar to the triple derivations of Lie algebras to some extent [33].…”

n-derivations of lie color algebras

“…Moreover, every derivation of the derivation algebra is an inner derivation. Double derivations of Lie superalgebras were introduced in [4], these derivations are similar to the triple derivations of Lie algebras to some extent in [7]. Zhao studied N-derivations of Lie algebras in [6].…”