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

TL;DR

Let $K$ be algebraically closed of characteristic zero and let $W_n(K)$ be the Lie algebra of polynomial vector fields on $K^n$. The paper constructs a family of maximal subalgebras $m_s(K)$ for $s=1, olinebreak olinebreak olinebreak olinebreak olinebreak olinebreak n-1$, proves their maximality and pairwise nonisomorphism, and describes their canonical ideal $I_s$ with $I_s$ isomorphic to $P_s$ tensor $Der(K[x_{s+1},...,x_n])$ and $m_s(K)/I_s$ isomorphic to $W_s(K)$. It further analyzes maximal subalgebras of rank $n$ over $P_n$, establishing a unique maximal polynomial ideal $M_0$ of rank $k$ contained in every rank$<n$ ideal and showing how $M/M_0$ has no ideals of rank $<n-k$; when $M$ is a polynomial subalgebra, it contains $I W_n$ for some ideal $I$ in $P_n$. Collectively, these results advance the classification and structural understanding of maximal subalgebras of $W_n(K)$ and reveal a clear decomposition pattern in terms of polynomial ideals and quotients by $I_s$.

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.

Theorems & Definitions (12)

• Lemma 1
• Lemma 2
• proof
• Theorem 1
• proof : Proof of (1)
• proof : Proof of (2)
• proof : Proof of (3)
• proof : Proof of (4)
• Remark 1
• Remark 2