On hereditary categories of quiver representations

We endeavor to study the representation theory of path algebras. Representation theory, in other words, is the study of abstract symmetries, which inform us on the nature of objects, even those which are intangible. A representation of an object gives us information about its structure much like a picture gives us information about a physical object manifesting in front of us. The objects we aim to study via their representations are called path algebras. These objects are sets equipped with an addition and a multiplication not unlike square matrices, which one can add and multiply, although the multiplication is not commutative. That said, path algebras are comprised of combinations of paths, and the multiplication on such a path algebra is defined via the concatentation of paths. Indeed, these objects are significant because they generalize many classically studied objects. For example, polynomials can be regarded as a path algebra. Upper triangular matrices can also be regarded as a path algebra. Of course, the scope of path algebras extends far beyond these two familiar objects. Thus, many of our results hold in a broad context. In fact, we show that a broad class of path algebras are hereditary. To say that a path algebra is hereditary is actually a conclusion about all of its representations, or its symmetries, that provides us greater control in comparing its representations to representations of other objects. It so happens we are able to leverage specific instances of this control in our work. In particular, for a slightly smaller class of path algebras we are afforded an invariant, which completely distinguishes the representations of a path algebra in that class up to equivalence. We compute several examples of this invariant, and in turn find, perhaps, a surprising example of two algebras with equivalent representation theories. We also use one instance of this invariant to draw a conclusion about a broad class of smaller (finite dimensional) algebras.