Nonlinear Lie type centralizers and derivations by local actions on triangular algebras

Abstract

Let $\mathcal{U}=Tri(\mathcal{A},\mathcal{M}, \mathcal{B})$ be a triangular algebra and $\mathcal{M}$ be a faithful $(\mathcal{A}, \mathcal{B})$-bimodule satisfying $\pi_{\mathcal{A}}(\mathcal{Z(U)})=\mathcal{Z(A)}$ and $\pi_{\mathcal{B}}(\mathcal{Z(U)}) = \mathcal{Z(B)}$. Suppose that a nonlinear map $\delta:\mathcal{U}\rightarrow \mathcal{U}$ satisfies $$\delta(p_{n(U_{1},U_{2},U_{3},...,U_{n}))=p_{n(\delta(U_{1}),U_{2},U_{3},...,U_{n})$$  and
$$\delta(p_{n}(U_{1},U_{2},U_{3},...,U_{n}))=p_{n(U_{1},\delta(U_{2}),U_{3},...,U_{n})$$ for all $U_{1},U_{2},...,U_{n}\in\mathcal{U}$ with $U_{1}U_{2}U_{3}=0.$ Then $\delta(U)=ZU+\tau(U)$ for some $Z\in\mathcal{Z(U)}$ and $\tau:\mathcal{U}\rightarrow \mathcal{Z(U)}$ is a map vanishing on each $(n-1)$-th commutator $p_{n}(U_{1},U_{2},U_{3},...,U_{n})$ with $U_{1}U_{2}U_{3}=0.$ Besides, we characterize nonlinear Lie type derivations by local actions on triangular algebras under certain conditions.
Suppose that a nonlinear map $\delta:\mathcal{U}\rightarrow \mathcal{U}$ satisfies $$\delta(p_{n(U_{1},U_{2},U_{3},...,U_{n}))=\sum_{i=1}^{n}p_{n}(U_{1},...,U_{i-1},\delta(U_{i}),U_{i+1},...,U_{n})$$
for all $U_{1},U_{2},...,U_{n}\in\mathcal{U}$ with $U_{1}U_{2}U_{3}=0$. Then $\delta(U)=d(U)+\tau(U)$ for all $U\in\mathcal{U}$, where $d:\mathcal{U}\rightarrow \mathcal{U}$ ia an additive derivation and $\tau:\mathcal{U}\rightarrow \mathcal{Z(U)}$ is a map vanishing on each $(n-1)$-th commutator $p_{n}(U_{1},U_{2},U_{3},...,U_{n})$ with $U_{1}U_{2}U_{3}=0.$