| Title:
|
On the nilpotent residuals of all subalgebras of Lie algebras (English) |
| Author:
|
Meng, Wei |
| Author:
|
Yao, Hailou |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
817-828 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $S(L)=L$. (English) |
| Keyword:
|
solvable Lie algebra |
| Keyword:
|
nilpotent residual |
| Keyword:
|
Frattini ideal |
| MSC:
|
17B05 |
| MSC:
|
17B20 |
| MSC:
|
17B30 |
| MSC:
|
17B50 |
| idZBL:
|
Zbl 06986974 |
| idMR:
|
MR3851893 |
| DOI:
|
10.21136/CMJ.2018.0006-17 |
| . |
| Date available:
|
2018-08-09T13:15:06Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147370 |
| . |
| Reference:
|
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