Maximal subalgebras of the Lie algebra $W_n(\mathbb{K})$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911707074396160 |
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| author | Chapovskyi, Y. Petravchuk, A. Tyshchenko, O. |
| author_facet | Chapovskyi, Y. Petravchuk, A. Tyshchenko, O. |
| contents | Let $K$ be an algebraically closed field of characteristic zero, $A= K[x_1, \dots, x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A.$ This Lie algebra also is the free $A$-module of rank $n$ over the ring $A,$ so every subalgebra of $W_n(K)$ has a rank $\leq n$ over $A.$ We prove that every maximal subalgebra of rank $\leq n$ of $W_n(K)$ is a simple Lie algebra. If a maximal subalgebra $L\subset W_n(K)$ has rank $n$ and is a submodule of $W_n(K)$ then $L$ is not simple. Moreover, $L$ is of the form $L=\{ D\in W_n(K) \ | \ D(I)\subseteq I\} $ for some ideal $I$ of the ring $A.$ It is also proved that, for a simple derivation $D$ on the ring $K[x, y]$, the subalgebra $K[x, y]D$ is a maximal subalgebra of $W_2(K).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximal subalgebras of the Lie algebra $W_n(\mathbb{K})$ Chapovskyi, Y. Petravchuk, A. Tyshchenko, O. Rings and Algebras 17B65, 17B66, 17B05 Let $K$ be an algebraically closed field of characteristic zero, $A= K[x_1, \dots, x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A.$ This Lie algebra also is the free $A$-module of rank $n$ over the ring $A,$ so every subalgebra of $W_n(K)$ has a rank $\leq n$ over $A.$ We prove that every maximal subalgebra of rank $\leq n$ of $W_n(K)$ is a simple Lie algebra. If a maximal subalgebra $L\subset W_n(K)$ has rank $n$ and is a submodule of $W_n(K)$ then $L$ is not simple. Moreover, $L$ is of the form $L=\{ D\in W_n(K) \ | \ D(I)\subseteq I\} $ for some ideal $I$ of the ring $A.$ It is also proved that, for a simple derivation $D$ on the ring $K[x, y]$, the subalgebra $K[x, y]D$ is a maximal subalgebra of $W_2(K).$ |
| title | Maximal subalgebras of the Lie algebra $W_n(\mathbb{K})$ |
| topic | Rings and Algebras 17B65, 17B66, 17B05 |
| url | https://arxiv.org/abs/2605.22970 |