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.
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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. |
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A similar example to Mariano's is the algebra generated by x,y subject to the relation $xy=1$. In a finite dimensional representation, yx must of course map to the same element as 1. However, there is a faithful infinite dimensional rep where x,y go to a unilateral shift and its adjoint so yx is not 1. |
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