Kronecker Quiver

In graph theory, 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  of a quiver assigns a vector space  to each vertex  of the quiver and a linear map  to each arrow .

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 to . 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 consists of two functions $m_v: V\to V'$ and $m_e: E\to E'$ such that the following diagrams commute:

That is,
:$m_v \circ s = s' \circ m_e$
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 s \\ 6pt\rightrightarrows \\ 4ptt \end\; 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 ... ''a''₃ ''a''₂ ''a''_''1'' such that the head of ''a''_''i''+1 is the tail of ''a''_''i'' for ''i'' = 1, ..., ''n''−1, using the convention of concatenating paths from right to left.

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'' 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 ''K''Γ are naturally identified with the representations of Γ. If the quiver has infinitely many vertices, then ''K''Γ has an approximate identity given by $e_F:=\sum_ 1_v$ 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).

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 for all vertices ''x'' in ''Q''.

A ''morphism'', , 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 ''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 ''K''Γ be its path algebra. Let ''e''_''i'' denote the trivial path at vertex ''i''. Then we can associate to the vertex ''i'' the projective ''K''Γ-module ''K''Γ''e_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 with relations

To enforce commutativity of some squares inside a quiver a generalization is the notion of quivers with relations (also named bound quivers).
A relation on a quiver is a linear combination of paths from .
A quiver with relation is a pair with a quiver and $I \subseteq K\Gamma$ an
ideal of the path algebra. The quotient is the path algebra of .

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, and consider stability conditions. These give quiver varieties, as constructed by .

Gabriel's theorem

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: , , , , .
# 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.

See also

* ADE classification
* Adhesive category
* Graph algebra
* Group ring
* Incidence algebra
* Quiver diagram
* Semi-invariant of a quiver
* Toric variety
*Derived noncommutative algebraic geometry - Quivers help encode the data of derived noncommutative schemes

References

Books

Lecture Notes

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* Quiver representations in toric geometry

Research

* Projective toric varieties as fine moduli spaces of quiver representations

Sources

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Errata

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*Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter functors, and Gabriel's theorem" (Russian), ''Uspekhi Mat. Nauk'' 28 (1973), no. 2(170), 19–33
Translation on Bernstein's website

* {{nlab, id=quiver, title=Quiver

Category theory
Representation theory
Directed graphs