A family of maximal subalgebras of the Lie algebra~$W_n(K)$
Тип публікації :
Препринт
Дата випуску :
23 лютого 2026 р.
Автор(и) :
Y. Chapovskyi
A. Petravchuk
eKNUTSHIR URL :
Журнал :
arXiv (Cornell University)
Цитування :
[APA 7] Y., C., & A., P. (2026). A family of maximal subalgebras of the Lie algebra~$W_n(K)$. arXiv (Cornell University),. https://ir.library.knu.ua/handle/15071834/34257
[ДСТУ] Y. C., A. P. A family of maximal subalgebras of the Lie algebra~$W_n(K)$. arXiv (Cornell University). 2026. URL: https://ir.library.knu.ua/handle/15071834/34257 (дата звернення: 11.09.2026).
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.
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