Weyl algebra ($A_n(k)$)

The first Weyl algebra is $A_1(k)=k⟨x,y⟩/(yx-xy-1)$, the algebra of polynomial-coefficient differential operators in one variable.

For the relative differential operators over $\mathbb{C}$ on a polynomial ring, coordinates have order zero and partial derivatives have order one. The associated graded ring is $\mathbb{C}[x,\xi ]$ with degrees zero for $x_i$ and one for ${\xi }_i$. This differs from the Bernstein filtration, in which both sets of generators have degree one and every filtration piece is finite-dimensional.

In a nonzero two-sided ideal choose an element of least Bernstein filtration degree. Commuting it with any position or differentiation generator lowers degree, so all such commutators vanish. The ordered monomial basis of a Weyl algebra and characteristic zero then force it to be a nonzero scalar. The ideal is the whole ring. The argument fails in positive characteristic, when powers of the generators can be central.

The defining commutators $[{\partial }_i,x_j]={\delta }_{ij}$ reorder every word into the displayed monomials. Over a characteristic-zero field, the polynomial representation proves independence: iterated commutation with the coordinate multiplications extracts a highest derivative coefficient, multiplied by the nonzero factorial $\beta !$. Hence the representation is faithful and every polynomial-coefficient operator has a unique ordered expression.

In characteristic $p>0$, multiplication by $x_i$ and formal differentiation by $x_i$ act on the displayed vector space, because differentiation preserves the ideal generated by the $p$th powers. They satisfy the Weyl algebra relations and yield a nonzero module of dimension $p^n$. The central elements $x_i^p$ and ${\partial }_i^p$ also show why the characteristic-zero simplicity argument cannot apply. In particular the characteristic-zero Bernstein inequality for Weyl algebra modules fails for this finite-dimensional module.

// Destination: module-theory.bigb

For the complex Weyl algebra $A_n$ with $n\ge 1$, a unital finite-dimensional module would give matrices satisfying $[{\partial }_1,x_1]=I$. Their matrix trace gives $0={\dim }_{\mathbb{C}}M$, so the module is zero. Characteristic zero and $n\ge 1$ are essential.

Give both position and differentiation generators of a Weyl algebra degree one. The ordered monomial basis gives $\dim F_j{A}_n=\left(\genfrac{}{}{0ex}{}{j+2n}{2n}\right)$ and ${\mathrm{gr}}_F{A}_n$ equal to a polynomial ring in $2n$ variables. Commuting a filtered element with any generator lowers its degree by at least one.