D-module - Introduction: Modules Over The Weyl Algebra

Introduction: Modules Over The Weyl Algebra

The first case of algebraic D-modules are modules over the Weyl algebra A_n(K) over a field K of characteristic zero. It is the algebra consisting of polynomials in the following variables

x₁, ..., x_n, ∂₁, ..., ∂_n.

where all of the variables x_i and ∂_j commute with each other, but the commutator

= ∂_ix_i − x_i∂_i = 1.

For any polynomial f(x₁, ..., x_n), this implies the relation

= ∂f / ∂x_i,

thereby relating the Weyl algebra to differential equations.

An (algebraic) D-module is, by definition, a left module over the ring A_n(K). Examples for D-modules include the Weyl algebra itself (acting on itself by left multiplication), the (commutative) polynomial ring K, where x_i acts by multiplication and ∂_j acts by partial differentiation with respect to x_j and, in a similar vein, the ring of holomorphic functions on Cn, the complex plane.

Given some differential operator P = a_n(x) ∂n + ... + a₁(x) ∂1 + a₀(x), where x is a complex variable, a_i(x) are polynomials, the quotient module M = A₁(C)/A₁(C)P is closely linked to space of solutions of the differential equation

P f = 0,

where f is some holomorphic function in C, say. The vector space consisting of the solutions of that equation is given by the space of homomorphisms of D-modules .