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.



Example of an algebra finite over a commutative subalgebra with infinite dimensional simple modules
Doc, this is a stinker. Your condition (2) forces your algebra to be finitely generated PI, and every little hare knows that simple modules over such algebras are finite-dimensional. See 13.4.9 and 13.10.3 of McConnell-Robson...


