This document discusses representation theory of quivers, which are directed graphs where a representation assigns a vector space to each vertex and linear maps between spaces for each arrow. It explores classifying quiver representations up to equivalence and connections to problems in linear algebra, Lie algebras, and quantum groups. In particular, it will study matrix problems related to Kronecker quivers and the correspondence between representation dimensions and positive roots of Lie algebras.

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Representation Theory of Quivers: Approach from Linear Algebra and Abstract Algebra
A quiver is a directed graph. A representation of a quiver is a collection of vector spaces, one for each
vertex, and linear maps between them, one for each arrow. A fundamental problem in this theory is to
study representations of quivers up to natural equivalences, given by base changes. This is a
generalization of classification problems from Linear Algebra (normal forms of matrices). It has
become clear that a whole range of problems of linear algebra can be formulated in a uniform way in
the context of representations of quivers. Even more, there are many surprising connections between
representation theory of quivers and other areas of mathematics, including Lie algebras and quantum
group. The starting point is Gabriel's Theorem, which states a beautiful relationship between
representations of quivers and root systems of Lie algebras. This connection has flourished into an
extensive theory including the subjects of Hall algebras and quiver varieties. By using the language of
quivers and their representations, we will study Gabriel-Roiter Matrix Problems, Similarity Matrix
Problems and Linking Matrix Problems. In particular, we will describe the classification of a pair of
matrices, which are related to Kronecker quivers. The one-to-one correspondence between dimension
vectors of indecomposable representations and positive roots of Lie algebras will be addressed. In the
setting of quantum group, by using the Hall algebra approach, we will study the decomposition of root
vectors as monomials of simple vectors. We will use the online open source software—“SageMath”
to do all symbolic calculations.