Generalized derivation extensions of 3-Lie algebras and corresponding Nambu–Poisson structures

Cohomologies and generalized derivation extensions of $n$-Lie algebras

“…In the present section we shall present the cohomological classification of generalized derivations of a -Lie algebras. In the case of = 2, the result recovers the analogue result recalled briefly in Subsection 1.1, while in the case of = 3 it answers the question raised in [17,Rk. 3.7].…”

“…Let us begin with the definition of a (left) Leibniz algebra following [17]. For the right handed version, we refer the reader to [14].…”

“…As for the condition (2.10), we have the following analogue of [17,Lemma 3.5]. Let us, however, record first the semi-direct sum Leibniz algebra L ⋊ L −1 := L ⊕ L −1 associated to a -Lie algebra L.…”

“…The case of derivation extensions, on the other hand, has been considered in [17] for 3-Lie algebras, and it is argued that a derivation of a 3-Lie algebra does not fit to define a 3-Lie algebra structure on the 1-dimensional extension ⊕ . The authors, therefore, introduced the notion of a generalized derivation for a 3-Lie algebra, which has later been extended to -Hom-Lie algebras in [18].…”

“…are in oneto-one correspondence with the 1st cohomology group of this algebraic object in question, with coefficients in itself. However, as is pointed out in [17] for 3-Lie algebras, such a characterization is missing for the case of generalized derivations. It is this gap that the present paper aims to fill in.…”