On the nilpotent residuals of all subalgebras of Lie algebras

摘要

  Let $mathcal{N}$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $mathbb{F}$, there exists a smallest ideal $I$ of $L$ such that $L/Iinmathcal{N}$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{mathcal{N}}$. In this paper, we define the subalgebra $S(L)=igcapolimits_{Hleq L}I_L(H^{mathcal{N}})$. Set $S_0(L) = 0$. Define $S_{i+1}(L)/S_i (L) =S(L/S_i (L))$ for $i geq1$. By $S_{infty}(L)$ denote the terminal term of the ascending series. It is proved that $L= S_{infty}(L)$ if and only if $L^{mathcal{N}}$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra $L$ with $S(L)=L$.