Characterizations of additive $\xi$-Lie derivations on unital algebras

Abstract

UDC 512.5

Let $\mathscr{R}$ be a commutative ring with unity and $\mathscr{U}$ be a unital algebra over $\mathscr{R}$ (or field $\mathbb{F}$).
An $\mathscr{R}$-linear map $L:\mathscr{U}\rightarrow\mathscr{U}$ is called a Lie derivation on $\mathscr{U}$ if $L([u,v])=[L(u),v]+[u,L(v)]$ holds for all $u,$ $v \in\mathscr{U}.$ For scalar $\xi\in\mathbb{F},$ an additive map $L\colon \mathscr{U}\rightarrow\mathscr{U}$ is called an additive $\xi$-Lie derivation on $\mathscr{U}$ if $L([u,v]_{\xi})=[L(u),v]_{\xi}+[u,L(v)]_{\xi},$ where $[u,v]_{\xi}=uv-\xi vu$ holds for all $u, v\in\mathscr{U}.$ In the present paper, under certain assumptions on $\mathscr{U}$
it is shown that every Lie derivation (resp., additive $\xi$-Lie derivation) ${L}$ on $\mathscr{U}$ is of standard form, i.e., $L=\delta+\phi,$ where $\delta$ is an additive derivation on $\mathscr{U}$ and $\phi$ is a mapping $\phi\colon \mathscr{U}\rightarrow Z(\mathscr{U})$ vanishing at $[u,v]$ with $uv=0$ in $\mathscr{U}.$ Moreover, we also characterize the additive $\xi$-Lie derivation for $\xi\neq 1$ by its action at zero product in a unital algebra over $\mathbb{F}.$