On the nilpotent residuals of all subalgebras of Lie algebras

Meng, Wei ; Yao, Hailou

Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 817-828

Résumé

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 $S(L)=L$.

Détail

DOI : 10.21136/CMJ.2018.0006-17

Classification : 17B05, 17B20, 17B30, 17B50
Keywords: solvable Lie algebra; nilpotent residual; Frattini ideal