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 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.