**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

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.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (5)

Related concepts (11)

Related courses (2)

Related lectures (2)

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).

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 construction of various types of generating functions used in combinatorics and number theory. A locally finite poset is one in which every closed interval [a, b] = {x : a ≤ x ≤ b} is finite.

MATH-473: Complex manifolds

The goal of this course is to help students learn the basic theory of complex manifolds and Hodge theory.

MATH-334: Representation theory

Study the basics of representation theory of groups and associative algebras.

Linear Algebra: Matrix Representation

Explores linear applications in R² and matrix representation, including basis, operations, and geometric interpretation of transformations.

Steenrod Squares

Covers the concept of Steenrod Squares and their applications in stable cohomology operations.

We compute the motivic Donaldson-Thomas invariants of the one-loop quiver, with an arbitrary potential. This is the first computation of motivic Donaldson-Thomas invariants to use in an essential way the full machinery of (mu) over cap -equivariant motives ...

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 ...

A fundamental problem in signal processing is to design computationally efficient algorithms to filter signals. In many applications, the signals to filter lie on a sphere. Meaningful examples of data of this kind are weather data on the Earth, or images o ...

2019