Weyl algebra

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 $k[x][y; \id, d/dx]$.