Бағытталған көпқырлы графтар және өрнектеу теориясы

Directed graph which is also a multigraph

In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows between two vertices are allowed. Quivers are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a. In category theory, a quiver can be understood to be the underlying structure of a category, but without composition or a designation of identity morphisms. That is, there is a forgetful functor from 'Cat' (the category of categories) to 'Quiv' (the category of multidigraphs). Its left adjoint is a free functor which, from a quiver, makes the corresponding free category.

A representation of a quiver Q is an association of an R module to each vertex of Q, and a morphism between each module for each arrow. A representation V of a quiver Q is said to be trivial if for all vertices x in Q. A morphism, f:V \to V', between representations of the quiver Q, is a collection of linear maps f(x):V(x) \to V'(x) such that for every arrow a in Q from x to y, i. e. the squares that f forms with the arrows of V and V' all commute. A morphism, f, is an isomorphism, if f (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 K Γ be its path algebra. Let ei denote the trivial path at vertex i. Then we can associate to the vertex i the projective K Γ module K Γei 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. This theory was related to cluster algebras by Derksen, Weyman, and Zelevinsky.

Given the dimensions of the vector spaces assigned to every vertex, one can form a variety which characterizes all representations of that quiver with those specified dimensions, and consider stability conditions. These give quiver varieties, as constructed by .

A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that:

A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the ADE Dynkin diagrams: An, Dn, E6, E7, E8. The indecomposable representations are in a one to one correspondence with the positive roots of the root system of the Dynkin diagram. found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite dimensional semisimple Lie algebras occur. This was generalized to all quivers and their corresponding Kac–Moody algebras by Victor Kac.