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W_n(K)的极大子代数族

A family of maximal subalgebras of the Lie algebra~$W_n(K)$

Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）；Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv（基辅塔拉斯舍甫琴科国立大学力学与数学系）

Y. Chapovskyi, A. Petravchuk

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AI总结：

本文研究了W_n(K)的极大子代数结构，证明了特定形式的子代数是极大子代数，并探讨了其理想与商代数的同构关系。

摘要：

设K为特征为零的代数闭域，P_n=K[x_1,…,x_n]为多项式环。任何K-导出D在P_n上都可以表示为D=∑_{i=1}^n f_i(x_1,…,x_n)∂/∂x_i，其中f_i∈P_n。所有这样的导出构成李代数W_n(K)。我们证明对于s=1,…,n-1，子代数m_s(K)={∑_{i=1}^s f_i∂/∂x_i +∑_{j=s+1}^n g_j∂/∂x_j | f_i∈P_s, g_j∈P_n}是W_n(K)的极大子代数。m_s(K)的理想I_s={∑_{j=s+1}^n g_j∂/∂x_j}同构于李代数P_s⊗Der(K[x_{s+1},…,x_n])，且m_s(K)/I_s≈W_s(K)。W_n(K)也是P_n上的自由模。因此，对于任意集合S⊆W_n(K)，其秩rk(S)（以P_n为系数环）被定义。指出W_n(K)中秩为n的极大子代数的一些性质。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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