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

TL;DR

This paper characterizes certain maximal subalgebras of the Lie algebra of polynomial derivations over an algebraically closed field, revealing their structure and relationships to other well-known Lie algebras.

Contribution

It introduces a family of maximal subalgebras of the Lie algebra $W_n(K)$ and describes their structure and properties, including their ideals and quotient relations.

Findings

The subalgebras $m_s(K)$ are maximal in $W_n(K)$ for $s=1, obreak ext{ to } obreak n-1$.

The ideal $I_s$ within $m_s(K)$ is isomorphic to a tensor product of polynomial functions and derivations.

The quotient $m_s(K)/I_s$ is isomorphic to $W_s(K)$, linking these subalgebras to smaller-dimensional Lie algebras.

Abstract

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.