Module structure of the Lie algebra $W_n(K)$ over $sl_n(K)$

Abstract

Let $\mathbb K$ be an algebraically closed field of characteristic zero, $A = \mathbb K[x_1,\dots,x_n]$ the polynomial ring, and let $W_n(\mathbb K)$ denote the Lie algebra of all $\mathbb K$-derivations on $A$. The Lie algebra $W_n := W_n(\mathbb K)$ admits a natural grading $W_n = \bigoplus_{i \ge -1} W^{[i]}_n$, where $W^{[i]}_n$ consists of all homogeneous derivations whose coefficients are homogeneous polynomials of degree $i+1$ or zero. The component $W^{[0]}_n$ is a subalgebra of $W_n$ and is isomorphic to $\mathfrak{gl}_n(\mathbb K).$ Moreover, each $W_n^{[i]}$ for $i \ge -1$ is a finite-dimensional module over $W_n^{[0]}$. We prove that $W^{[i]}_n,\; i \ge 0$ is a sum of two irreducible submodules $W^{[i]}_n = M_i \oplus N_i$, where $M_i$ consists of all divergence-free derivations, and $N_i$ consists of derivations that are polynomial multiples of the Euler derivation $E_n = \sum_{i=1}^n x_i \frac{\partial}{\partial x_i}$. As a consequence, we show that the standard grading is exact in certain sense, namely: $[W^{[i]}_n, W^{[j]}_n] = W^{[i+j]}_n$ for all $i,j,$ except when $i = j = 0$. We also address the question of when the subalgebra of $W_n$ generated by $W_n^{[-1]} \oplus W_n^{[0]},$ together with an additional element from $W_n,$ equals the entire Lie algebra $W_n$.