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

Abstract

Let $$$K$$$ be an algebraically closed field of characteristic zero and $$${P_n=K[x_1,\ldots,x_n]}$$$ be 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 $$$W_n:=W_n(K)$$$ 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 $$$W_n$$$. 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 $$$W_n(K)$$$ admits a natural grading $$$W_n = \bigoplus_{i \ge -1} W^{[i]}_n$$$. For the subalgebra $$$L=W_n^{[-1]}\oplus W_n^{[0]}$$$ and any $$$D\in W_n$$$ some conditions on $$$D$$$ are pointed out under which $$$L$$$ and $$$D$$$ generate the entire algebra $$$W_n$$$.