  Several classes of *-algebras associated to the action of an affine
transformation are considered, and an investigation of the interplay between
the different classes of algebras is initiated. Connections are established
that relate representations of *-algebras, geometry of algebraic surfaces,
dynamics of affine transformations, graphs and algebras coming from a
quantization procedure of Poisson structures. In particular, algebras related
to surfaces being inverse images of fourth order polynomials (in R^3) are
studied in detail, and a close link between representation theory and geometric
properties is established for compact as well as non-compact surfaces.
