Non-global nonlinear Lie $n$-derivations on unital algebras with idempotents

Abstract

Let $\mathfrak{T}$ be a unital algebra with nontrivial idempotents. For any $s_1, s_2,\cdots, s_n\in\mathfrak{T},$ \linebreak define $p_1(s_1)=s_1$, $p_2(s_1, s_2)=[s_1, s_2]$ and $p_n(s_1, s_2,\cdots, s_n)=[p_{n-1}(s_1, s_2,\cdots, s_{n-1}), s_n]$ for all integers $n\geq 3.$ In the present article, it is shown that if a map
$\varphi:\mathfrak{T}\rightarrow \mathfrak{T}$ satisfies
$$\varphi(p_n(s_1, s_2, \cdots, s_n))=\sum_{i=1}^{n} p_n(s_1,\cdots, s_{i-1}, \varphi(s_i), s_{i+1}, \cdots, s_n)\ \ (n\geq 3)$$ for all $s_1, s_2, \cdots, s_n\in\mathfrak{T}$ with $s_1s_2\cdots s_n=0,$ then $\varphi(s+t)-\varphi(s)-\varphi(t)\in \mathcal{Z}(\mathfrak{T})$ for all $s, t\in\mathfrak{T},$ and under some mild assumptions $\varphi$ is of the form $\delta+\tau,$ where $\delta:\mathfrak{T}\rightarrow\mathfrak{T}$ is an additive derivation and $\tau:\mathfrak{T}\rightarrow \mathcal{Z}(\mathfrak{T})$ is a map such that $\tau(p_n(s_1, s_2, \cdots, s_n))=0$ for all $s_1, s_2, \cdots, s_n\in\mathfrak{T}$ with $s_1s_2\cdots s_n=0.$ The above results are then applied to certain special classes of unital algebras, namely triangular algebras, full matrix algebras and algebra of all bounded linear operators.