Quiver (mathematics)

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 Template:Mvar of a quiver assigns a vector space Template:Math to each vertex Template:Mvar of the quiver and a linear map Template:Math to each arrow Template:Mvar.

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 Template:Math (the category of categories) to Template:Math (the category of multidigraphs). Its left adjoint is a free functor which, from a quiver, makes the corresponding free category.

Definition

A quiver Template:Math consists of:

• The set Template:Mvar of vertices of Template:Math
• The set Template:Mvar of edges of Template:Math
• Two functions: Template:Tmath giving the start or source of the edge, and another function, Template:Tmath giving the target of the edge.

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. Formally, if <math>\Gamma=(V,E,s,t)</math> and <math>\Gamma'=(V',E',s',t')</math> are two quivers, then a morphism <math>m=(m_v, m_e)</math> of quivers consists of two functions <math>m_v: V\to V'</math> and <math>m_e: E\to E'</math> such that the following diagrams commute:

That is,

<math>m_v \circ s = s' \circ m_e</math>

and

<math>m_v \circ t = t' \circ m_e</math>

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) Template:Mvar is a category with two objects, and four morphisms: The objects are Template:Mvar and Template:Mvar. The four morphisms are Template:Tmath Template:Tmath and the identity morphisms Template:Tmath and Template:Tmath That is, the free quiver is the category

<math>E

\;\begin{matrix} s \\[-6pt] \rightrightarrows \\[-4pt] t \end{matrix}\; V</math>

A quiver is then a functor Template:Tmath. (That is to say, <math>\Gamma</math> specifies two sets <math>\Gamma(V)</math> and <math>\Gamma(E)</math>, and two functions <math>\Gamma(s),\Gamma(t)\colon \Gamma(E) \longrightarrow \Gamma(V)</math>; this is the full extent of what it means to be a functor from <math>Q</math> to <math>\mathbf{Set}</math>.)

More generally, a quiver in a category Template:Mvar is a functor Template:Tmath The category Template:Math of quivers in Template:Mvar is the functor category where:

• objects are functors Template:Tmath
• morphisms are natural transformations between functors.

Note that Template:Math is the category of presheaves on the opposite category Template:Math.

Path algebra

If Template:Math is a quiver, then a path in Template:Math is a sequence of arrows

<math>a_n a_{n-1} \dots a_3 a_2 a_1</math>

such that the head of Template:Math is the tail of Template:Mvar for Template:Math, 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 Template:Mvar is a field then the quiver algebra or path algebra Template:Math is defined as a vector space having all the paths (of length ≥ 0) in the quiver as basis (including, for each vertex Template:Mvar of the quiver Template:Math, a trivial path Template:Mvar of length 0; these paths are not assumed to be equal for different Template:Mvar), 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 Template:Mvar. This algebra has a unit element if and only if the quiver has only finitely many vertices. In this case, the modules over Template:Math are naturally identified with the representations of Template:Math. If the quiver has infinitely many vertices, then Template:Math has an approximate identity given by <math display="inline">e_F:=\sum_{v\in F} 1_v</math> where Template:Mvar ranges over finite subsets of the vertex set of Template:Math.

If the quiver has finitely many vertices and arrows, and the end vertex and starting vertex of any path are always distinct (i.e. Template:Mvar has no oriented cycles), then Template:Math is a finite-dimensional hereditary algebra over Template:Mvar. Conversely, if Template:Mvar is algebraically closed, then any finite-dimensional, hereditary, associative algebra over Template:Mvar 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 Template:Mvar is an association of an Template:Mvar-module to each vertex of Template:Mvar, and a morphism between each module for each arrow.

A representation Template:Mvar of a quiver Template:Mvar is said to be trivial if <math>V(x)=0</math> for all vertices Template:Mvar in Template:Mvar.

A morphism, Template:Tmath between representations of the quiver Template:Mvar, is a collection of linear maps Template:Tmath such that for every arrow Template:Mvar in Template:Mvar from Template:Mvar to Template:Mvar, <math>V'(a)f(x) = f(y)V(a),</math> i.e. the squares that Template:Mvar forms with the arrows of Template:Mvar and Template:Mvar all commute. A morphism, Template:Mvar, is an isomorphism, if Template:Math is invertible for all vertices Template:Mvar in the quiver. With these definitions the representations of a quiver form a category.

If Template:Mvar and Template:Mvar are representations of a quiver Template:Mvar, then the direct sum of these representations, <math>V\oplus W,</math> is defined by <math>(V\oplus W)(x)=V(x)\oplus W(x)</math> for all vertices Template:Mvar in Template:Mvar and <math>(V\oplus W)(a)</math> is the direct sum of the linear mappings Template:Math and Template:Math.

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 Template:Mvar is just a covariant functor from this category to the category of finite dimensional vector spaces. Morphisms of representations of Template:Mvar are precisely natural transformations between the corresponding functors.

For a finite quiver Template:Math (a quiver with finitely many vertices and edges), let Template:Math be its path algebra. Let Template:Mvar denote the trivial path at vertex Template:Mvar. Then we can associate to the vertex Template:Mvar the projective Template:Math-module Template:Math consisting of linear combinations of paths which have starting vertex Template:Mvar. This corresponds to the representation of Template:Math obtained by putting a copy of Template:Mvar at each vertex which lies on a path starting at Template:Mvar and 0 on each other vertex. To each edge joining two copies of Template:Mvar we associate the identity map.

This theory was related to cluster algebras by Derksen, Weyman, and Zelevinsky.<ref>Template:Citation. Published in J. Amer. Math. Soc. 23 (2010), p. 749-790.</ref>

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 Template:Mvar is a Template:Mvar linear combination of paths from Template:Mvar. A quiver with relation is a pair Template:Math with Template:Mvar a quiver and <math>I \subseteq K\Gamma</math> an ideal of the path algebra. The quotient Template:Math is the path algebra of Template:Math.

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 Template:Harvtxt.

Gabriel's theorem

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

1. 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: Template:Math.
2. The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

Template:Harvtxt 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.

See also

• ADE classification
• Adhesive category
• Assembly theory
• 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

Lecture Notes

• Template:Citation
• Quiver representations in toric geometry

Research

• Projective toric varieties as fine moduli spaces of quiver representations

Sources

• Template:Citation
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• Template:Citation
• Template:Citation.
• Victor Kac, "Root systems, representations of quivers and invariant theory". Invariant theory (Montecatini, 1982), pp. 74–108, Lecture Notes in Math. 996, Springer-Verlag, Berlin 1983. Template:ISBN<ref>Template:Cite book</ref>
• Template:Citation
• Template:Citation
• Template:Citation
• 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.
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