Quiver (mathematics) - Representations of Quivers

Representations of Quivers

A representation V of a quiver Q is said to be trivial if V(x) = 0 for all vertices x in Q.

A morphism, ƒ : V → V', between representations of the quiver Q, is a collection of linear maps such that for every arrow a in Q from x to y, i.e. the squares that ƒ forms with the arrows of V and V' all commute. A morphism, ƒ, is an isomorphism, if ƒ(x) is invertible for all vertices x in the quiver. With these definitions the representations of a quiver form a category.

If V and W are representations of a quiver Q, then the direct sum of these representations, is defined by for all vertices x in Q and is the direct sum of the linear mappings V(a) and W(a).

A representation is said to be decomposable if it is isomorphic to the direct sum of non-zero representations.

A categorical definition of a quiver representation can also be given. The quiver itself can be considered a category, where the vertices are objects and paths are morphisms. Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces. Morphisms of representations of Q are precisely natural transformations between the corresponding functors.

For a finite quiver Γ (a quiver with finitely many vertices and edges), let KΓ be its path algebra. Let e_i denote the trivial path at vertex i. Then we can associate to the vertex i, the projective KΓ module KΓe_i consisting of linear combinations of paths which have starting vertex i. This corresponds to the representation of Γ obtained by putting a copy of K at each vertex which lies on a path starting at i and 0 on each other vertex. To each edge joining two copies of K we associate the identity map.