Quivers and Representations

Abstract

Publisher Summary This chapter discusses the use of the language of quivers in mathematics. Quivers arise in many areas of mathematics, including representation theory, algebraic geometry and differential geometry, Kac–Moody algebras, and quantum groups. A quiver is just a directed graph, and a representation of a quiver associates a vector space to each vertex and a linear map to each arrow. The introduction of quivers marked the starting point of the modern representation theory of finite-dimensional associative algebras. A number of remarkable connections to other mathematical fields have been discovered, in particular to Hall algebras, quantum groups, elliptic Lie algebras, and more recently cluster algebras. The representation theory of finite-dimensional algebras can be formulated using the language of quivers. In effect, the study of representations of a finite-dimensional algebra over an algebraically closed field is reduced to the study of a quotient of path algebra of a certain quiver. If the base field is not algebraically closed, “species” (or valued graphs) instead of quivers can be used to realize finite-dimensional algebras.