Quiver (mathematics)

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They 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 an underlying structure of a category, but without identity morphisms and composition. That is, there is a forgetful functor from Cat to Quiv. Its left adjoint is a free functor which, from a quiver, makes the corresponding free category.

Definition

A quiver Γ consists of:

The set V of vertices of Γ
The set E of edges of Γ
Two functions: s: E → V giving the start or source of the edge, and another function, t: E → V giving the target of the edge.

This definition is identical to that of a multidigraph.

A morphism of quivers is defined as follows. If $\Gamma =(V,E,s,t)$ and $\Gamma '=(V',E',s',t')$ are two quivers, then a morphism $m=(m_{v},m_{e})$ of quivers consist of two functions $m_{v}:V\to V'$ and $m_{e}:E\to E'$ such that following diagrams commute:

m_{v}\circ s=s'\circ m_{e}

and

m_{v}\circ t=t'\circ m_{e}

Category-theoretic definition

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.

The free quiver (also called the walking quiver, Kronecker quiver, 2-Kronecker quiver or Kronecker category) Q is a category with two objects, and four morphisms: The objects are V and E. The four morphisms are s: E → V, t: E → V, and the identity morphisms id_V: V → V and id_E: E → E. That is, the free quiver is

E\;{\begin{matrix}s\\[-6pt]\rightrightarrows \\[-4pt]t\end{matrix}}\;V

A quiver is then a functor Γ: Q → Set.

More generally, a quiver in a category C is a functor Γ: Q → C. The category Quiv(C) of quivers in C is the functor category where:

objects are functors Γ: Q → C,
morphisms are natural transformations between functors.

Note that Quiv is the category of presheaves on the opposite category Q^op.

Path algebra

If Γ is a quiver, then a path in Γ is a sequence of arrows a_n a_n−1 ... a₃ a₂ a₁ such that the head of a_i+1 = tail of a_i, using the convention of concatenating paths from right to left.

If K is a field then the quiver algebra or path algebra KΓ 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 $e_{i}$ of length 0; these paths are not assumed to be equal for different i), and multiplication given by concatenation of paths. If two paths cannot be concatenated because the end vertex of the first is not equal to the starting vertex of the second, their product is defined to be zero. This defines an associative algebra over K. This algebra has a unit element if and only if the quiver has only finitely many vertices. In this case, the modules over KΓ are naturally identified with the representations of Γ. If the quiver has infinitely many vertices, then KΓ has an approximate identity given by $e_{E}:=\sum _{v\in E}1_{v}$ where E 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 KΓ is a finite-dimensional hereditary algebra over K and conversely any such finite-dimensional hereditary algebra over K is isomorphic to the path algebra over its Ext quiver

Representations of quivers

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 V(x) = 0 for all vertices x in Q.

A morphism, f: V → V′, between representations of the quiver Q, is a collection of linear maps $f(x):V(x)\rightarrow V'(x)$ such that for every arrow a in Q from x to y $V'(a)f(x)=f(y)V(a)$ , 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, $V\oplus W$ , is defined by $(V\oplus W)(x)=V(x)\oplus W(x)$ for all vertices x in Q and $(V\oplus W)(a)$ 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.

Quiver Variety

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. These give Nakajima quiver varieties.^[1]

Gabriel's theorem

A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. Gabriel (1972) 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: $A_{n}$ , $D_{n}$ , $E_{6}$ , $E_{7}$ , $E_{8}$ .
The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

Dlab & Ringel (1973) found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite dimensional semisimple Lie algebras occur.

References

^ Ginzburg, Victor. Lectures on Nakajima's Quiver Varieties

Derksen, Harm; Weyman, Jerzy (February 2005), "Quiver Representations" (PDF), Notices of the American Mathematical Society, 52 (2)
Dlab, Vlastimil; Ringel, Claus Michael (1973), On algebras of finite representation type, Carleton Mathematical Lecture Notes, vol. 2, Department of Mathematics, Carleton Univ., Ottawa, Ont., MR 0347907
Crawley-Boevey, William (1992), Notes on Quiver Representations (PDF), Oxford University
Gabriel, Peter (1972), "Unzerlegbare Darstellungen. I", Manuscripta Mathematica, 6 (1): 71–103, doi:10.1007/BF01298413, ISSN 0025-2611, MR 0332887. Errata.
Alistair, Savage (2006), "Finite-dimensional algebras and quivers", in Francoise, J.-P.; Naber, G. L.; Tsou, S.T. (eds.), Encyclopedia of Mathematical Physics, vol. 2, Elsevier, pp. 313–320, arXiv:math/0505082
Simson, Daniel; Skowronski, Andrzej; Assem, Ibrahim (2007), Elements of the Representation Theory of Associative Algebras, Cambridge University Press, ISBN 978-0-521-88218-7
Quiver at the nLab

[1] Ginzburg, Victor. Lectures on Nakajima's Quiver Varieties

[1]