A237/2003 - Simple Shamsuddin Derivations of K[X_1,...,X_n] and cyclic maximal left ideals of the Weyl algebra A

Preprint A237/2003

Simple Shamsuddin Derivations of K[X_1,...,X_n] and cyclic maximal left ideals of the Weyl algebra A_n(K)

Yves LEQUAIN

Keywords: simple derivation | maximal left ideal | Weyl algebra | holonomic module

Let K be a field of characteristic zero and A_n the nth-Weyl algebra over K. Let d be a derivation of the type d = d/dX_1 + f_2.d/dX_2 + ... + f_n.d/dX_n, where f_i = a_iX_i + b_i with a_i and b_i elements of K[X_1] for every i= 2,...,n. First, we show that one can determine effectively whether d is a simple derivation of K[X_1,...,X_n] or not. Next, we characterize the polynomials h of the type h = f_2 + ... + f_n with f_i an element of K[X_1,X_i] for every i=2,...,n, such that d+h generates a maximal left ideal of k[X_1,...,X_n] or not, hence also such that A_n/A_n.(d+h) is an holonomic module over A_n or not. Finally, we use our algorithm to establish important new families of non-holonomic modules over A_n.