---
tag: 98QE7KY55W
kind: concept
name: Weyl algebra
msc: [16S32, 16S36]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl_algebra
same_as:
  wikidata: Q1069378
  wikipedia: Weyl algebra
---

## Summary

The $n$-th Weyl algebra $A_n(k)$ over a field $k$ is the algebra generated by $x_1, \dots, x_n$ and $y_1, \dots, y_n$ with the relations $y_i x_i - x_i y_i = 1$.
All other pairs of generators commute.
In characteristic zero it is the [[ring of differential operators|GBWQF3HZG0]] with polynomial coefficients, where $x_i$ acts by multiplication and $y_i$ by $\partial / \partial x_i$.
In characteristic zero it is the basic example of a simple Noetherian domain that is not a division ring.
It also models the canonical commutation relation of quantum mechanics.

## Explanations

### L1 human

Take polynomials in $x$ and the operator $D = d/dx$ that differentiates.
By the product rule, $D(x f) = x D(f) + f$, so the operators satisfy $D x - x D = 1$.
The first Weyl algebra is the algebra built from two symbols with exactly this rule.
Every element can be written uniquely as a finite sum $\sum a_{ij} x^i D^j$, like a polynomial in two variables that do not commute.