No! The Weyl algebra is simple. The problem of classifying the finite dimensional representations is wild. This was discussed in
http://mathoverflow.net/questions/18559/

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.