Abstract
Transcript of Abstract
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.
1