On the transitivity of Lie ideals and a characterization of perfect Lie algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Souris, Nikolaos Panagiotis |
| author_facet | Souris, Nikolaos Panagiotis |
| contents | We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being a Lie ideal, in the sense that if a Lie algebra $\mathfrak{h}$ is a subideal of a Lie algebra $\mathfrak{g}$ (i.e. there exist Lie subalgebras $\mathfrak{l}_0,\mathfrak{l}_1,\dots,\mathfrak{l}_n$ of $\mathfrak{g}$ with $\mathfrak{h}=\mathfrak{l}_0\unlhd \mathfrak{l}_1 \unlhd\cdots \unlhd \mathfrak{l}_n=\mathfrak{g}$), then $\mathfrak{h}$ is an ideal of $\mathfrak{g}$. We also prove that perfect Lie algebras of arbitrary dimension and over any field are intrinsically characterized by transitivity of this type; In particular, we show that a Lie algebra $\mathfrak{h}$ is perfect (i.e. $\mathfrak{h}=[\mathfrak{h}, \mathfrak{h}]$) if and only if for any Lie algebra $\mathfrak{g}$ such that $\mathfrak{h}$ is a subideal of $\mathfrak{g}$, it follows that $\mathfrak{h}$ is an ideal of $\mathfrak{g}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the transitivity of Lie ideals and a characterization of perfect Lie algebras Souris, Nikolaos Panagiotis Rings and Algebras 17B05 (Primary), 17B40, 22E60 (Secondary) We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being a Lie ideal, in the sense that if a Lie algebra $\mathfrak{h}$ is a subideal of a Lie algebra $\mathfrak{g}$ (i.e. there exist Lie subalgebras $\mathfrak{l}_0,\mathfrak{l}_1,\dots,\mathfrak{l}_n$ of $\mathfrak{g}$ with $\mathfrak{h}=\mathfrak{l}_0\unlhd \mathfrak{l}_1 \unlhd\cdots \unlhd \mathfrak{l}_n=\mathfrak{g}$), then $\mathfrak{h}$ is an ideal of $\mathfrak{g}$. We also prove that perfect Lie algebras of arbitrary dimension and over any field are intrinsically characterized by transitivity of this type; In particular, we show that a Lie algebra $\mathfrak{h}$ is perfect (i.e. $\mathfrak{h}=[\mathfrak{h}, \mathfrak{h}]$) if and only if for any Lie algebra $\mathfrak{g}$ such that $\mathfrak{h}$ is a subideal of $\mathfrak{g}$, it follows that $\mathfrak{h}$ is an ideal of $\mathfrak{g}$. |
| title | On the transitivity of Lie ideals and a characterization of perfect Lie algebras |
| topic | Rings and Algebras 17B05 (Primary), 17B40, 22E60 (Secondary) |
| url | https://arxiv.org/abs/2405.20893 |