In
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, a quiver is a
directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...
where
loops and multiple arrows between two
vertices are allowed, i.e. a
multidigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by mor ...
. They are commonly used in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
: a representation of a quiver assigns a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
to each vertex of the quiver and a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
to each arrow .
In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a quiver can be understood to be the underlying structure of a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
, but without composition or a designation of identity morphisms. That is, there is a
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
from to . Its
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
is a
free functor
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between elem ...
which, from a quiver, makes the corresponding
free category In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.
More precisely, the objec ...
.
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
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by mor ...
.
A
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
of quivers is defined as follows. If
and
are two quivers, then a morphism
of quivers consists of two functions
and
such that the following
diagrams commute:
![Quiver Morphism Target Diagram](https://upload.wikimedia.org/wikipedia/commons/5/56/Quiver_Morphism_Target_Diagram.svg)
That is,
:
and
:
Category-theoretic definition
The above definition is based in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
; the category-theoretic definition generalizes this into a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from the ''free quiver'' to the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
.
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 morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s id
''V'': ''V'' → ''V'' and id
''E'': ''E'' → ''E''. That is, the free quiver is
:
A quiver is then a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
where:
* objects are functors Γ: ''Q'' → ''C'',
* morphisms are
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s between functors.
Note that Quiv is the
category of presheaves
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on ...
on the
opposite category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
''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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
over ''K''. This algebra has a unit element if and only if the quiver has only finitely many vertices. In this case, the
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
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
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
Definition
A right approximate ...
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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
hereditary algebra In mathematics, especially in the area of abstract algebra known as module theory, a ring ''R'' is called hereditary if all submodules of projective modules over ''R'' are again projective. If this is required only for finitely generated submodu ...
over ''K''. Conversely, if ''K'' is algebraically closed, then any finite-dimensional, hereditary, associative algebra over ''K'' is
Morita equivalent
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...
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
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
.
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
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from this category to the category of finite dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s. Morphisms of representations of ''Q'' are precisely
natural transformations
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
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
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
Ade, Adé, or ADE may refer to:
Aeronautics
*Ada Air's ICAO code
* Aden International Airport's IATA code
*Aeronautical Development Establishment, a laboratory of the DRDO in India
Medical
* Adverse Drug Event
*Antibody-dependent enhancement
* A ...
Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
s: , , , , .
# The indecomposable representations are in a one-to-one correspondence with the positive roots of the
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
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
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
*
Adhesive category
In mathematics, an adhesive category is a category where pushouts of monomorphisms exist and work more or less as they do in the category of sets. An example of an adhesive category is the category of directed multigraphs, or quivers, and the the ...
*
Graph algebra
In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, and has seen many uses in the field of universa ...
*
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
*
Incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural constructi ...
*
Quiver diagram
In theoretical physics, a quiver diagram is a graph representing the matter content of a gauge theory that describes D-branes on orbifolds. Quiver diagrams may also be used to described \mathcal = 2 supersymmetric gauge theories in four dimens ...
*
Semi-invariant of a quiver
In mathematics, given a Quiver (mathematics), quiver Q with set of vertices Q0 and set of arrows Q1, a representation (mathematics), representation of Q assigns a vector space ''V'i'' to each vertex and a linear map ''V''(''α''): ''V''(''s''('' ...
*
Toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be nor ...
*
Derived noncommutative algebraic geometry In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some bas ...
- Quivers help encode the data of derived noncommutative schemes
References
Books
Lecture Notes
*
*
Quiver representations in toric geometry
Research
*
Projective toric varieties as fine moduli spaces of quiver representations
Sources
*
*
*
*
Errata
*
*
*
*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