On the nilpotent residuals of all subalgebras of Lie algebras

@article{Meng2018,
abstract = {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 $S(L)=\bigcap \nolimits _\{H\le L\}I_L(H^\{\mathcal \{N\}\})$. Set $S_0(L) = 0$. Define $S_\{i+1\}(L)/S_i (L) =S(L/S_i (L))$ for $i \ge 1$. 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 $S(L)=L$.},
author = {Meng, Wei, Yao, Hailou},
journal = {Czechoslovak Mathematical Journal},
keywords = {solvable Lie algebra; nilpotent residual; Frattini ideal},
language = {eng},
number = {3},
pages = {817-828},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the nilpotent residuals of all subalgebras of Lie algebras},
url = {http://eudml.org/doc/294446},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Meng, Wei
AU - Yao, Hailou
TI - On the nilpotent residuals of all subalgebras of Lie algebras
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 817
EP - 828
AB - 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 $S(L)=\bigcap \nolimits _{H\le L}I_L(H^{\mathcal {N}})$. Set $S_0(L) = 0$. Define $S_{i+1}(L)/S_i (L) =S(L/S_i (L))$ for $i \ge 1$. 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 $S(L)=L$.
LA - eng
KW - solvable Lie algebra; nilpotent residual; Frattini ideal
UR - http://eudml.org/doc/294446
ER -