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.

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