Weyl algebra

In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.

In the simplest case, these are differential operators. Let ${\displaystyle F}$ be a field, and let ${\displaystyle F[x]}$ be the ring of polynomials in one variable with coefficients in ${\displaystyle F}$. Then the corresponding Weyl algebra consists of differential operators of form

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

In this case, ${\displaystyle q}$ corresponds to left multiplication by ${\displaystyle x}$, and ${\displaystyle p}$ corresponds to taking derivative with respect to ${\displaystyle x}$. This is the first Weyl algebra ${\displaystyle A_{1}}$. The n-th Weyl algebra ${\displaystyle A_{n}}$ are constructed similarly.

Alternatively, ${\displaystyle A_{1}}$ can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by ${\displaystyle ([p,q]-1)}$. Similarly, ${\displaystyle A_{n}}$ is obtained by quotienting the free algebra on 2n generators by the ideal generated by$${\displaystyle ([p_{i},q_{j}]-\delta _{i,j}),\quad \forall i,j=1,\dots ,n}$$where ${\displaystyle \delta _{i,j}}$ is the Kronecker delta.

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 case of ${\displaystyle A_{n}}$ with underlying field ${\displaystyle F}$ characteristic zero, unless otherwise stated.

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.