On n-Derivations and n-Homomorphisms in Perfect Lie Superalgebras

Let (Formula presented.) be a fixed integer. The aim of this paper is to investigate the properties of n-derivations within the framework of perfect Lie superalgebras over a commutative ring R. The main result shows that if the base ring contains (Formula presented.), and L is a perfect Lie superalgebra with a center equal to zero, then any n-derivation of L is necessarily a derivation. Additionally, every n-derivation of the derivation algebra (Formula presented.) is an inner derivation. Moreover, we extend the concept of n-homomorphisms to mappings between Lie superalgebras L and (Formula presented.) and prove that under specific assumptions, homomorphisms, anti-homomorphisms, and their combinations are all n-homomorphisms. Finally, we conclude our paper with some open problems.