Module structure of the Lie algebra $W_n(K)$ over $sl_n(K)$
Y. Chapovskyi, A. Petravchuk
TL;DR
Let $W_n = W_n(\mathbb{K})$ be the Lie algebra of polynomial vector fields on $\mathbb{K}^n$ with grading $W_n=\bigoplus_{i\ge -1}W_n^{[i]}$ and $W_n^{[0]}\cong \mathfrak{gl}_n(\mathbb{K})$. The authors show that each nonnegative graded component $W_n^{[i]}$ splits as $M_i\oplus N_i$, where $M_i$ consists of divergence-free derivations and $N_i$ consists of derivations that are polynomial multiples of the Euler derivation $E_n$, with $E_n=\sum_{t=1}^n x_t\frac{\partial}{\partial x_t}$. Both $M_i$ and $N_i$ are irreducible $W_n^{[0]}$-modules, and the brackets satisfy $[W_n^{[i]},W_n^{[j]}]=W_n^{[i+j]}$ for all $i,j\ge 0$, except when $i=j=0$, where $W_n^{[0]}=M_0\oplus N_0$ with $M_0\simeq \mathfrak{sl}_n(\mathbb{K})$. The paper further derives explicit dimension formulas, establishes detailed Bracket relations among the components, and provides a criterion for when a single extra derivation, together with $W_n^{[-1]}\oplus W_n^{[0]}$, generates the entire $W_n$, illuminating the maximal-subalgebra structure of this Lie algebra.
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 $W_n$.