On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$

TL;DR

This paper proves that certain subalgebras of the Lie algebra of derivations on polynomial rings are maximal with respect to being locally nilpotent or solvable, highlighting their algebraic significance.

Contribution

It establishes the maximality of the triangular subalgebra as locally nilpotent and the subalgebra $s_n(K)$ as solvable within $W_n(K)$.

Findings

$u_n(K)$ is a maximal locally nilpotent subalgebra.

$s_n(K)$ is a maximal solvable subalgebra.

Derived length of $s_n(K)$ is $2n$.

Abstract

Let $K$ be an algebraically closed field of characteristic zero, $P_n=K[x_1, ..., x_n]$ the polynomial ring, and $W_n(K)$ the Lie algebra of all $K$-derivations on $P_n$. One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K) = P_0\partial_1+\cdots+P_{n-1}\partial_n$, where $\partial_i:=\partial/\partial x_i$ are partial derivatives on $P_n$. This subalgebra consists of locally nilpotent derivations on $P_n.$ Such derivations define automorphisms of the ring $P_n$ and were studied by many authors. The subalgebra $u_n(K) $ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1P_0)\partial_1+\cdots +(P_{n-1}+x_nP_{n-1})\partial_n,$ which is solvable of the derived length $2n$ that is the maximum derived length of solvable subalgebras of $W_n(K).$ It is proved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K).$