This chapter collects a number of important examples of modules over the Weyl algebra. The prototype of all the examples we discuss here is the polynomial ring in n variables; and with it we shall begin. The reader is expected to be familiar with the basic notions of module theory, as explained in [Cohn, Ch.10].

THE POLYNOMIAL RING.

In Ch. 1, the Weyl algebra was constructed as a subring of an endomorphism ring. Writing K[X] for the polynomial ring K[x1, …, xn] we have that An(K) is a subring of EndKK[X]. One deduces from this that the polynomial ring is a left An-module. Thus the action of xi on K[X] is by straightforward multiplication; whilst ∂i acts by differentiation with respect to xn. This is a very important example, and we shall study it in some detail. Let us first recall some basic definitions.

Let us first recall some basic definitions. Let R be a ring. An R-module is irreducible, or simple, if it has no proper submodules. Let M be a left R-module. An element u ∈ M is a torsion element if annR(u) is a non-zero left ideal. If every element of M is torsion, then M is called a torsion module.