Quiver (mathematics)
In mathematics, a designated quiver (English quiver ) a directed graph , d. That is , a quiver consists of a set of points and a set of arrows as well as two images that assign each arrow its starting point (s for source ) and its target point (t for target ).
The designation of a directed graph as a quiver is only common in representation theory .
Depiction of a quiver
In representation theory, a representation of a quiver consists of a family of vector spaces and a family of vector space homomorphisms. The vector spaces should be those over a fixed body .
A morphism , between two representations of a quiver is a family of linear maps so that each arrow of according applies: .
With the help of these definitions, the representations of a quiver form a category . In this a morphism is an isomorphism if and only if it is invertible for every point of the quiver .
example
Representation of a quiver with two vector spaces and a vector space homomorphism .
properties
The undirected graph on which the quiver is based is denoted by (i.e. clearly simple: the arrows are made into edges). A quiver is called connected if the underlying undirected graph is connected.
A representation of a quiver is said to be decomposable if it is either trivial (i.e. consists only of zero vector spaces and zero morphisms ) or if it can be written as the direct sum of two non-trivial sub-representations. Otherwise the representation is called indecomposable.
A quiver is of a finite representation type if, apart from isomorphism, it only has a finite number of indivisible representations.
Theorem by Gabriel
A connected quiver is of finite representation type if and only if a Dynkin diagram is of type or ( Pierre Gabriel 1972).
Foreigner riding theory
To a finite-dimensional - algebra over a field a so-called can Auslander-Reiten-quiver be defined, with the points of the quiver, the isomorphism classes of indecomposable modules of algebra and the arrows so-called irreducible figures are between the modules. The Auslands riding theory finally introduces methods of homology theory into the representation theory of quivers.
literature
- Maurice Auslander , Idun riding , Smalo: Representation Theory of Artin Algebras. Cambridge University Press, 1995. ISBN 978-0521411349
- Ibrahim Assem et al .: Elements of representation theory of associative algebras . Cambridge University Press, 2006. ISBN 0-521-58423-X
- Harm Derksen , Jerzy Weyman: An Introduction to Quiver Representations , Vol. 184, American Mathematical Society, 2017, ISBN 978-1-4704-2556-2 .