$\mathcal{D}$-modules are very important objects of algebraic analysis and algebraic geometry. We briefly talked about it in the last post. $\mathcal{D}$-modules are modules over the ring of differential operators. In this post, let us briefly revise the Weyl Algebra because it is very important to enter the world of $\mathcal{D}$-modules. Weyl Algebra can be constructed as a $\mathcal{D}$- module.
The best application of $\mathcal{D}$-modules has been to solve the differential equations with Bernard Malgrange applied to the (differential) equations with constant coefficients initially. Then, Kashiwara in 1971 applied the equations with analytic coefficients. And if I am right, this was also the second birth of algebraic analysis after Sato.
And as said, Coutinho, $\mathcal{D}$- modules have two branches which depend on the base variety: algebraic and analytic. Moreover, $\mathcal{D}$- modules are not restricted to the Weyl Algerbas.
Let $K$ be a field of characteristic zero and $K[X]$ be the ring of polynomials. Now, $K[X]$ is an infinite-dimensional vector space over K, and the linear operators are denoted by the $End_K K[X]$. These will be two linear operators $x_i$ and $\partial_i$. (Remember that polynomials in $K[X]$ are defined in $n$ commuting indeterminates over $K$). Let us take a polynomial $m$ in $K[X]$. The operators are defined as
$$x_i (m) = x_i \cdot m$$
$$ \partial_i(m) = \frac{\partial m}{\partial x_i}$$
The Weyl Algebra $A_n$ is defined as the subalgebra of $End_K K[X]$ (and hence a $K$-algebra) generated from $x_i$ ($x_1,x_2,\cdots,x_n$) and $\partial_i$ ($\partial_1,\partial_2 \cdots, \partial_n$).
Moreover, the most important part of Weyl Algebra is that they are not commutative. So one finds,
$$\partial_i\cdot x_i = x_i \cdot \partial_i +1$$
and hence $[\partial_i, x_i]=1$ (it is very easy to see this).
A Historical Note: Weyl Algebra played a crucial role in the development of quantum mechanics (see this). Also, $A_n$ notation was used by Dixmier to denote the algebra that physicists use to describe systems with $n$ degrees of freedom.
