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

Abstract

Let $$$K$$$ be an algebraically closed field of characteristic zero,  $$$P_n=K[x_1,\ldots ,x_n]$$$  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 $$$P_n$$$ and $$$P_0=K.$$$ 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 $$$W_n(K)$$$.