A family of maximal subalgebras of the Lie algebra~$W_n(K)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914343924269056 |
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| author | Chapovskyi, Y. Petravchuk, A. |
| author_facet | Chapovskyi, Y. Petravchuk, A. |
| contents | Let $K$ be an algebraically closed field of characteristic zero and ${P_n=K[x_1,\ldots,x_n]}$ the polynomial ring. Any $K$-derivation $D$ on $P_n$ is of the form ${ D=\sum_{i=1}^n f_i(x_1,\ldots,x_n)\frac{\partial}{\partial x_i} },$ where $f_i\in P_n.$ All such derivations form the Lie algebra $W_n(K)$ over the field $K$. We prove that for $s=1,\ldots,n-1$ the subalgebra $
m_s(K)=\left\{ \sum_{i=1}^s f_i\frac{\partial}{\partial x_i} +\sum_{j=s+1}^n g_j\frac{\partial}{\partial x_j} \mid f_i\in P_s,\ g_j\in P_n \right\} $
is a maximal subalgebra of~$W_n(K)$. The ideal $
I_s=\left\{\sum_{j=s+1}^n g_j\frac{\partial}{\partial x_j}\right\} $
of $m_s(K)$ is isomorphic to the Lie algebra $P_s\otimes \mathrm{Der}(K[x_{s+1},\ldots,x_n])$ and $m_s(K)/I_s\simeq W_s(K)$. The Lie algebra $W_n(K)$ is also the free module over the ring $P_n.$ Therefore, for any set $S\subseteq W_n(K)$ the rank $rk(S)$ (over $P_n$) is defined. Some properties of maximal subalgebras of rank $n$ in $W_n(K)$ are pointed out. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19601 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A family of maximal subalgebras of the Lie algebra~$W_n(K)$ Chapovskyi, Y. Petravchuk, A. Rings and Algebras 17B65, 17B66, 17B05 Let $K$ be an algebraically closed field of characteristic zero and ${P_n=K[x_1,\ldots,x_n]}$ the polynomial ring. Any $K$-derivation $D$ on $P_n$ is of the form ${ D=\sum_{i=1}^n f_i(x_1,\ldots,x_n)\frac{\partial}{\partial x_i} },$ where $f_i\in P_n.$ All such derivations form the Lie algebra $W_n(K)$ over the field $K$. We prove that for $s=1,\ldots,n-1$ the subalgebra $ m_s(K)=\left\{ \sum_{i=1}^s f_i\frac{\partial}{\partial x_i} +\sum_{j=s+1}^n g_j\frac{\partial}{\partial x_j} \mid f_i\in P_s,\ g_j\in P_n \right\} $ is a maximal subalgebra of~$W_n(K)$. The ideal $ I_s=\left\{\sum_{j=s+1}^n g_j\frac{\partial}{\partial x_j}\right\} $ of $m_s(K)$ is isomorphic to the Lie algebra $P_s\otimes \mathrm{Der}(K[x_{s+1},\ldots,x_n])$ and $m_s(K)/I_s\simeq W_s(K)$. The Lie algebra $W_n(K)$ is also the free module over the ring $P_n.$ Therefore, for any set $S\subseteq W_n(K)$ the rank $rk(S)$ (over $P_n$) is defined. Some properties of maximal subalgebras of rank $n$ in $W_n(K)$ are pointed out. |
| title | A family of maximal subalgebras of the Lie algebra~$W_n(K)$ |
| topic | Rings and Algebras 17B65, 17B66, 17B05 |
| url | https://arxiv.org/abs/2602.19601 |