Preprint A235/2003
Cyclic Maximal Left Ideals of the Weyl Algebra A_2(K): An Effective Approach.

Cydara RIPOLL | LEQUAIN , Yves | DOERING, Ada Maria

**Keywords: **
simple derivation; maximal ideal of the Weyl álgebra; holonomic module

Let K be a field of characteristic zero and A_2 the second Weyl algebra over K. Let d be a derivation of the type
d = d/dX + gd/dY, with g an element of K[X,Y], such that
K[X,Y] is d-simple. First, we show that there exists an element a of {1,-1} such that d+aY generates a maximal left ideal of A_2. Next, we show that if h is any element of
K[X,Y], then one can effectively determine whether d+h generates a maximal left ideal of A_2 or not, hence also whether A_2/A_2.(d+h) is an irreducible non-holonomic module over A_2 or not. Finally, as an application, we study the family of examples d = d/dX + (Y^2 - p(X))d/dY, where p(X) belongs to K[X] with deg(p(X)) of the type 6k+3 or 6k+5, where k is any integer bigger than or equal to 0.