Affine transformation crossed product like algebras and noncommutative surfaces

TL;DR

This paper explores *-algebras linked to affine transformations, examining their representations, geometric surfaces, and dynamics, revealing deep connections between algebraic, geometric, and dynamical systems.

Contribution

It introduces new classes of *-algebras associated with affine transformations and investigates their relationships with geometric surfaces and quantization of Poisson structures.

Findings

Connections between algebra representations and surface geometry

Detailed study of algebras related to polynomial inverse images

Link between representation theory and geometric properties

Abstract

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.