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Keywords:
solvable Lie algebra; nilpotent residual; Frattini ideal
Summary:
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 $L/I\in \mathcal {N}$. 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 $S(L)=\bigcap \nolimits _{H\leq L}I_L(H^{\mathcal {N}})$. Set $S_0(L) = 0$. Define $S_{i+1}(L)/S_i (L) =S(L/S_i (L))$ for $i \geq 1$. 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 $S(L)=L$.
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