Weyl algebra

The first Weyl algebra over a field $k$ isIn characteristic zero, it acts faithfully on $k[x]$: $x$ multiplies by the variable and $\partial$ differentiates. The defining relation is the product rule, since $\partial (xf)-x\partial (f)=f$. Every element has a unique ordered expression ${\sum }_{i,j}{a}_{ij}{x}^i{\partial }^j$, giving the ordered monomial basis of a Weyl algebra.

The higher Weyl algebra $A_n(k)$ has commuting coordinates $x_i$, commuting derivatives ${\partial }_i$, and $[{\partial }_i,x_j]={\delta }_{ij}$. It organizes polynomial-coefficient differential operators. The abstract algebra remains meaningful in positive characteristic, but its usual action on polynomials is no longer faithful: ${\partial }^p$ acts as zero in characteristic $p$ while remaining a nonzero ordered monomial. See Weyl algebra in positive characteristic for the resulting finite-dimensional representations.