The Trajectory Coset and Similarity Classes of Affine Maps

TL;DR

This paper introduces the trajectory coset, an algebraic tool to analyze the similarity classes of affine maps, bridging geometric properties with algebraic invariants to classify affine transformations.

Contribution

It develops the concept of the trajectory coset and uses it to determine similarity classes of affine maps through algebraic invariants.

Findings

Defined the trajectory coset for affine maps

Established algebraic invariants for similarity classification

Connected geometric properties with algebraic module isomorphisms

Abstract

In this work we define the trajectory coset of an affine map and use it to study the similarity classes of affine maps. We use the trajectory coset, a tool which allows us to gain a deeper understanding of the interplay between geometry (properties of affine maps) and algebra (properties of linear maps). We first state the geometric problem of similarity of affine maps. We then develop the algebraic tools. The main idea is the development of an invariant which determines whether one coset can be taken to another coset, under isomorphism of modules. After resolving this problem we go back to geometrical questions, similarity and invariant flats.