D-SIMPLE RINGS AND PRINCIPAL MAXIMAL IDEALS OF THE WEYL ALGEBRA

Abstract

We prove that if the order-one differential operator $S=\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i + \gamma$, with $\beta_i,\gamma \in K[x_1,\ldots,x_n]$, generates a maximal left ideal of the Weyl algebra $A_n(K)$, then $S$ does not admit any Darboux differential operator in $K[x_1,\ldots,x_n]\langle \partial_2,\ldots,\partial_n\rangle $; hence in particular, the derivation $\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i$ does not admit any Darboux polynomial in $K[x_1,\ldots,x_n]$. We show that the converse is true when $\beta_i \in K[x_1,x_i]$, for every $i=2,\ldots,n$. Then, we generalize to $K[x_1,\ldots,x_n]$ the classical result of Shamsuddin that characterizes the simple linear derivations of $K[x_1,x_2]$. Finally, we establish a criterion for the left ideal generated by $S$ in $A_n(K)$ to be maximal in terms of the existence of polynomial solutions of a finite system of differential polynomial equations.