No! The Weyl algebra is simple. The problem of classifying the finite dimensional representations is wild. This was discussed in Is there a machinery describing all the irreducible representations ?Is there a machinery describing all the irreducible representations ?
In case this is too succinct. The Weyl algebra is the algebra of linear differential operators with polynomial coefficients. It is generated by $x$ and $D$ with defining relation $Dx-xD=1$. This acts on polynomials in $x$ with $D=d/dx$. This is a faithful simple representation.
The problem of finding finite dimensional representations is the problem of solving linear differential equations.
