I'm interested in finding an algebra with elements x,y which are identified by every finite-dimensional module. I'm primarily interested in the case that everything is over the complex field, but answers over other fields would also be interesting.



Algebra with elements x, y such that r(x)=r(y) for all finite-dimensional modules r
The Weyl algebra, generated by $x$ and $y$ subject to the single relation $xy-yx=1$, is an example. A silly one, one may add: it does not have any finite dimensional representation!


There are more interesting examples: let $\mathfrak g$ be the Lie algebra with basis $x\_i$, $y\_i$, with $i\in\mathbb Z$, and $z$, such that the only non-zero brackets are $[x\_i,y\_i]=z$, for all $i\in\mathbb Z$. If $A=\mathcal U(\mathfrak g)$ is the enveloping algebra of $\mathfrak g$, then $z$ acts by zero in every finite dimensional module.


There are finitely-generated examples, too.


