---
tag: JJZ5X28NNR
kind: definition
name: Weyl algebra
msc: [16S32, 16S36]
defines: 98QE7KY55W
uses: [04C5ZBG1WT, 2P3Y2FRCSZ, 0CD87SFM2N, 0076HWHGHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl_algebra
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $n \ge 1$ be an integer.
Let $k\langle x_1, \dots, x_n, y_1, \dots, y_n \rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $2n$ generators.
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] generated by the elements
$$x_i x_j - x_j x_i, \qquad y_i y_j - y_j y_i, \qquad y_i x_j - x_j y_i - \delta_{ij}$$
for $1 \le i, j \le n$, where $\delta_{ij}$ is $1$ if $i = j$ and $0$ otherwise.
The *$n$-th Weyl algebra* is the [[quotient ring|0076HWHGHQ]] $A_n(k) = k\langle x_1, \dots, x_n, y_1, \dots, y_n \rangle / I$.

## Remarks

The monomials $x_1^{a_1} \cdots x_n^{a_n} y_1^{b_1} \cdots y_n^{b_n}$ with $a_i, b_i \ge 0$ form a basis of $A_n(k)$ over $k$.
Example: in $A_1(k)$, with $y x = x y + 1$, one gets $y x^2 = (x y + 1) x = x (y x) + x = x(xy + 1) + x = x^2 y + 2x$.
This matches the rule $\frac{d}{dx}(x^2 f) = x^2 f' + 2 x f$ for the operators of multiplication by $x$ and differentiation.
Non-example: the commutative polynomial ring $k[x, y]$ satisfies the first two kinds of relations but has $y x - x y = 0 \ne 1$.
If $k$ has characteristic $0$, then $A_n(k)$ is a simple Noetherian domain with center $k$.
If $k$ has characteristic $p > 0$, then $x_1^p$ lies in the center of $A_n(k)$, and $A_n(k)$ is not simple.
Equivalently, $A_1(k)$ is the Ore extension $k[x][y; \id, d/dx]$.