Cohomologies, deformations and extensions of n-Hom-Lie algebras

Abstract

In this paper, first we give the cohomologies of an n-Hom-Lie algebra and introduce the notion of a derivation of an n-Hom-Lie algebra. We show that a derivation of an n-Hom-Lie algebra is a 1-cocycle with the coefficient in the adjoint representation. We also give the formula of the dual representation of a representation of an n-Hom-Lie algebra. Then, we study (n − 1)-order deformation of an n-Hom-Lie algebra. We introduce the notion of a Hom-Nijenhuis operator, which could generate a trivial (n − 1)-order deformation of an n-Hom-Lie algebra. Finally, we introduce the notion of a generalized derivation of an n-Hom-Lie algebra, by which we can construct a new n-Hom-Lie algebra, which is called the generalized derivation extension of an n-Hom-Lie algebra.

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

“…In order to classify the derivation extensions of -Lie algebras, via cohomology, we shall now recall the cohomology theory for -Lie algebras, developed in [2], to a cohomology theory with arbitrary coefficients, see for instance [18,Sect. 3].…”

“…Then, in [2], this cohomology theory has been considered further, in close connection with the Leibniz cohomology. This cohomology theory, associated to -Lie algebras, upgraded recently to a cohomology theory with arbitrary coefficients in [1] and [20] in the case of = 3, and in [18] to a cohomology theory associated to a -Hom-Lie algebra along with a representation space of it.…”

“…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].…”

“…, −1 ∈ L, see[18, Def. 5.1].It follows at once, as is noted in[18, Lemma 5.3], that for any ∈ L ad :L −1 → L, ad ( 1 , . .…”

On the Cohomology and Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras

“…Abelian extensions and nonabelian extensions of Hom-Lie algebras are, respectively, researched in [21,22]. The general extensions of n-Hom-Lie algebras is researched in [23]. In 1997, Bordemann introduced the notion of T * -extensions of Lie algebras in [24].…”