Generalized derivations of n-Hom Lie superalgebras

TL;DR

This paper explores the structure of generalized derivations in multiplicative n-Hom Lie superalgebras, extending previous results and revealing how these derivations relate and embed within larger algebraic frameworks.

Contribution

It generalizes key results on derivations from n-Lie algebras to multiplicative n-Hom Lie superalgebras, including new properties and embeddings.

Findings

Quasiderivations can be embedded as derivations in larger superalgebras.

Derivations decompose as a direct sum when the center is zero.

Established hierarchy and properties of various derivations.

Abstract

It is well known that n-Hom Lie superalgebras are certain generalizations of n-Lie algebras. This paper is devoted to investigate the generalized derivations of multiplicative n-Hom Lie superalgebras. We generalize the main results of Leger and Luks to the case of multiplicative n-Hom Lie superalgebras. Firstly, we review some concepts associated with a multiplicative n-Hom Lie superalgebra $N$. Furthermore, we give the definitions of the generalized derivations, quasiderivations, center derivations, centroids and quasicentroids. Obviously, we have the following tower $ZDer(N)\subseteq Der(N)\subseteq QDer(N)\subseteq GDer(N)\subseteq End(N)$. Later on, we give some useful properties and connections between these derivations. Moreover, we obtain that the quasiderivation of $N$ can be embedded as a derivation in a larger multiplicative n-Hom Lie superalgebra. Finally, we conclude that the derivation of the larger multiplicative n-Hom Lie superalgebra has a direct sum decomposition when the center of $N$ is equal to zero.