Path algebra

Summary

The path algebra $kQ$ of a quiver $Q$ over a field $k$ is the vector space with basis the paths in $Q$. Its multiplication is concatenation of paths. For a quiver with finitely many vertices, its modules are the same as the representations of the quiver. Every finite-dimensional algebra over an algebraically closed field is Morita equivalent to a quotient of a path algebra by an ideal of relations. This makes path algebras the basic examples of representation theory of finite-dimensional algebras.