Weyl algebra has no nonzero finite-dimensional modules ($\mathrm{tr}[{\partial }_1,x_1]=0$)

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.