The goal of this talk is to introduce the quantum spin model and discuss the math structures required to analyze the model. We will begin by introducing an interpretation of the physics that gives rise to a quantum spin system and what it means for the model to evolve according to quantum mechanics. This will involve introducing the mathematical objects that are used to study quantum mechanics. Once we have all of the math structures in place, we want to extend our theory from the finite volume to the infinite volume. But in doing so, some of the mathematical structure is lost. We will discuss the challenges of extending dynamics to the infinite system and a way to recover the lost structures using some light representation theory.