quiver representations and representation morphisms
Let be a quiver, i.e. is a set of vertices, is a set of arrows and are functions such that maps each arrow to its source and maps each arrow to its target.
A morphism between representations and is a family of -linear maps such that for each arrow the following relation holds:
Obviously we can compose morphisms of representations and in this the case class of all representations and representation morphisms together with the standard composition is a category. This category is abelian.
It can be shown that for each finite quiver (i.e. with finite) and field there exists an algebra over such that the category of representations of is equivalent to the category of modules over .
A representation of is called trivial iff for each vertex .
A representation of is called locally finite-dimensional iff for each vertex and finite-dimensional iff is locally finite-dimensional and for almost all vertices .
|Title||quiver representations and representation morphisms|
|Date of creation||2013-03-22 19:16:15|
|Last modified on||2013-03-22 19:16:15|
|Last modified by||joking (16130)|