On the parameterization of primitive ideals in affine PI algebras

We consider the following question, concerning associative algebras R over an algebraically closed field k: When can the space of (equivalence classes of) finite dimensional irreducible representations of R be topologically embedded into a classical affine space? We provide an affirmative answer for algebraic quantum groups at roots of unity. More generally, we give an affirmative answer for k-affine maximal orders satisfying a polynomial identity, when k has characteristic zero. Our approach closely follows the foundational studies by Artin and Procesi on finite dimensional representations. Our results also depend on Procesi's later study of Cayley-Hamilton identities.