${\textit{n}}$th Pointwise Inner Derivation of $\textit{n}$-Isoclinism Lie Algebras

Abstract

Let L be a finite-dimensional Lie algebra and I,  J two ideals of L. Let $\mathrm {Der}_J^I(L)$ denote the set of all derivations of L whose images are in I and send J to zero and let $\mathrm {Der}^n_c(L)$ denote the set of all derivations $\alpha$ of L for which $\alpha (x)\in [x,L^n]$ for all $x\in L$. In this paper, we have shown that if L and H are two n-isoclinic Lie algebras, then there exists an isomorphism from $\mathrm {Der}^{L^{n+1}}_{Z_n(L)}(L)$ to $\mathrm {Der}^{H^{n+1}}_{Z_n(H)}(H)$. Also, we give necessary and sufficient conditions under which $\mathrm {Der}^n_c(L)$ is equal to a certain special subalgebra of the derivation algebra of L.