= Weyl algebra
{c}
{title2=$A_n(k)$}
{wiki}
The first <Weyl algebra> over a <field> $k$ is
$$
A_1(k)=k\langle x,\partial\rangle/(\partial x-x\partial-1).
$$
In <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.
Back to article page
