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 ). ${\displaystyle Q}$ ${\displaystyle Q_{0}}$ ${\displaystyle Q_{1}}$ ${\displaystyle s,t:Q_{1}\rightarrow Q_{0}}$

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 . ${\displaystyle Q}$ ${\displaystyle \left\{V(i):i\in Q_{0}\right\}}$ ${\displaystyle \left\{(V(a):V(i)\rightarrow V(j)):(a:i\rightarrow j)\in Q_{1}\right\}}$

A morphism , between two representations of a quiver is a family of linear maps so that each arrow of according applies: . ${\displaystyle f:V\rightarrow V'}$ ${\displaystyle Q}$ ${\displaystyle \left\{f(i):V(i)\rightarrow V'(i):i\in Q_{0}\right\}}$ ${\displaystyle a\in Q_{1}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle V'(a)f(i)=f(j)V(a)}$

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 . ${\displaystyle f=\left\{f(i):V(i)\rightarrow V'(i):i\in Q_{0}\right\}}$ ${\displaystyle f(i)}$ ${\displaystyle i}$

example

${\displaystyle V_{1}{\overset {f}{\longrightarrow }}V_{2}}$

Representation of a quiver with two vector spaces and a vector space homomorphism . ${\displaystyle V_{1},V_{2}}$ ${\displaystyle f\colon V_{1}\to V_{2}}$

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. ${\displaystyle \left|Q\right|}$ ${\displaystyle Q}$

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). ${\displaystyle Q}$ ${\displaystyle \left|Q\right|}$ ${\displaystyle A_{n},D_{n},E_{6},E_{7}}$ ${\displaystyle E_{8}}$

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. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$

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 .