Quiver (mathematics)
In mathematics, a quiver (English Quiver ) denotes a directed graph, that is, a quiver consists of a set of points and a set of arrows and two illustrations that each arrow its starting point ( s for source) and its target ( t assign for target).
The designation of a directed graph as a quiver is common only in the representation theory.
Representation of a quiver
In the representation theory is a representation of a quiver from a family of vector spaces and a family of Vektorraumhomomorphismen. The vector spaces should there be such a fixed chosen over a field.
A morphism between two representations of a quiver is a family of linear maps, such that for every arrow from to.
Using these definitions, the representations of a quiver form a category. This is a morphism is an isomorphism if it is invertible for each point of the quiver.
Example
Representation of a quiver with two vector spaces and a homomorphism of vector.
Properties
With the quiver underlying undirected graph is called (ie, vividly simple: making the arrows to edges). A quiver is called connected if the underlying undirected graph is connected.
A representation of a quiver is called separable, if it is either trivial ( ie only zero vector spaces and zero morphisms is ) or if it can be written as a direct sum of two non-trivial sub- representations. Otherwise, the representation is called indecomposable.
A quiver is of finite representation type if it has only finitely many indecomposable representations up to isomorphism.
Set of Gabriel
A quiver is of finite representation type if and only if a Dynkin diagram is of type or (Pierre Gabriel 1972).
To a finite-dimensional algebra over a field can be defined in a so-called Auslander - Reiten- quiver, the points of the quiver isomorphism classes of indecomposable modules of the algebra and the arrows are called irreducible maps between the modules. The Auslander - Reiten- theory eventually leads so that methods of homology theory in the representation theory of quivers one.