Let $A$ be an algebra over an algebraically closed field $k.$ Recall that if $A$ is a finitely generated module over its center, and if its center is a finitely generated algebra over $k,$ then by the Schur's lemma all simple $A$-modules are finite dimensional over $k.$
Motivated by the above, I would like an example of a $k$-algebra $A,$ such that:
1) $A$ has a simple module of infinitie dimension over $k,$
2) $A$ contains a commutative finitely generated subalgebra over which $A$ is a finitely generated left and right module.
Thanks in advance.