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.
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.
A quiver Γ consists of: This definition is identical to that of a multidigraph.
A morphism of quivers is a mapping from vertices to vertices which takes directed edges to directed edges.
That is, and The above definition is based in set theory; the category-theoretic definition generalizes this into a functor from the free quiver to the category of sets.
If Γ is a quiver, then a path in Γ is a sequence of arrows such that the head of ai+1 is the tail of ai for i = 1, …, n−1, using the convention of concatenating paths from right to left.
Note that a path in graph theory has a stricter definition, and that this concept instead coincides with what in graph theory is called a walk.
If K is a field then the quiver algebra or path algebra K Γ is defined as a vector space having all the paths (of length ≥ 0) in the quiver as basis (including, for each vertex i of the quiver Γ, a trivial path ei of length 0; these paths are not assumed to be equal for different i), and multiplication given by concatenation of paths.
If the quiver has infinitely many vertices, then K Γ has an approximate identity given by
where F ranges over finite subsets of the vertex set of Γ.
If the quiver has finitely many vertices and arrows, and the end vertex and starting vertex of any path are always distinct (i.e. Q has no oriented cycles), then K Γ is a finite-dimensional hereditary algebra over K. Conversely, if K is algebraically closed, then any finite-dimensional, hereditary, associative algebra over K is Morita equivalent to the path algebra of its Ext quiver (i.e., they have equivalent module categories).
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.
between representations of the quiver Q, is a collection of linear maps
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.
The quiver itself can be considered a category, where the vertices are objects and paths are morphisms.
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.
A relation on a quiver Q is a K linear combination of paths from Q.
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 King (1994).
Gabriel (1972) classified all quivers of finite type, and also their indecomposable representations.
More precisely, Gabriel's theorem states that: Dlab & Ringel (1973) 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.
Kirillov, Alexander (2016), Quiver Representations and Quiver Varieties, American Mathematical Society, ISBN 978-1-4704-2307-0