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Concept# Quiver (mathematics)

Summary

In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, in other words 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 , a quiver can be understood to be the underlying structure of a , but without composition or a designation of identity morphisms. 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 .
A quiver Γ consists of:
The set V of vertices of Γ
The set E of edges of Γ
Two functions: s:E \to V giving the start or source of the edge, and another function, t:E \to 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 and are two quivers, then a morphism of quivers consists of two functions and such that the following diagrams commute:
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 .
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 \to V, t: E \to V, and the identity morphisms \mathrm{id}_V: V \to V and \mathrm{id}_E: E \to E. That is, the free quiver is
A quiver is then a functor \Gamma:Q \to \mathbf{Set}.
More generally, a quiver in a category C is a functor \Gamma: Q \to C. The category Quiv(C) of quivers in C is the where:
objects are functors \Gamma:Q \to C,
morphisms are natural transformations between functors.
Note that Quiv is the on the Qop.
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.

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Related publications (7)

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Quiver (mathematics)

In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, in other words 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 , a quiver can be understood to be the underlying structure of a , but without composition or a designation of identity morphisms. That is, there is a forgetful functor from Cat to Quiv.

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. In formal terms, a directed graph is an ordered pair where V is a set whose elements are called vertices, nodes, or points; A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.

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 essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

Related lectures (2)

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the $p$-adic pushforward of the Haar measure under a hypertoric moment map $\mu$. This leads to an explicit formula for the Igusa zeta function $\FI_\mu(s)$ of $\mu$, and in particular to a small set of candidate poles for $\FI_\mu(s)$. We also study various properties of the residue at the largest pole of $\FI_\mu(s)$. Finally, if $\mu$ is constructed out of a quiver $\Gamma$ we give a conjectural description of this residue in terms of indecomposable representations of $\Gamma$ over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or $p$-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.

We prove that Hausel’s formula for the number of rational points of a Nakajima quiver variety over a finite field also holds in a suitable localization of the Grothendieck ring of varieties. In order to generalize the arithmetic harmonic analysis in his proof we use Grothendieck rings with exponentials as introduced by Cluckers-Loeser and Hrushovski-Kazhdan.

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