Weyl algebra

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form

${\displaystyle f_{m}(X)\partial _{X}^{m}+f_{m-1}(X)\partial _{X}^{m-1}+\cdots +f_{1}(X)\partial _{X}+f_{0}(X).}$

More precisely, let F be the underlying field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each f_i lies in F[X], ∂_X is the derivative with respect to X, and the algebra is generated by X and ∂_X. For example:$${\displaystyle (aX-1)\partial _{X}^{3}+(-X^{5}+b)\partial _{X}-2}$$where ${\displaystyle a,b\in F}$.

The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.

More generally, let ${\displaystyle (R,\Delta )}$ be a partial differential ring with commuting derivatives ${\displaystyle \Delta =\lbrace \partial _{1},\ldots ,\partial _{m}\rbrace }$. The Weyl algebra associated to ${\displaystyle (R,\Delta )}$ is the noncommutative ring ${\displaystyle R[\partial _{1},\ldots ,\partial _{m}]}$ satisfying the relations ${\displaystyle \partial _{i}r=r\partial _{i}+\partial _{i}(r)}$ for all ${\displaystyle r\in R}$. The previous case is the special case where ${\displaystyle R=F[x_{1},\ldots ,x_{n}]}$ and ${\displaystyle \Delta =\lbrace \partial _{x_{1}},\ldots ,\partial _{x_{n}}\rbrace }$ where ${\displaystyle F}$ is a field. This article discusses only the special case unless otherwise stated.