Representation of a quiver

Summary

A representation of a quiver over a field assigns a vector space to each vertex. It also assigns to each arrow a linear map from the space at its source to the space at its target. Representations of a finite quiver are the same as modules over its path algebra. Classifying the indecomposable representations is a central problem; Gabriel's theorem and Kac's theorem describe their dimension vectors through root systems. Many problems of linear algebra, such as the classification of pairs of matrices, are problems about quiver representations.