representations of a bound quiver
Let be a bound quiver (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) over a field .
Let be a representation of over composed by a family of linear maps. If
is a path in , then we have the evaluation map
For stationary paths we define by . Also, note that if is a relation (http://planetmath.org/RelationsInQuiver) in , then
where all ’s have the same source and target. Thus it makes sense to talk about evaluation in , i.e.
is a linear map.
Recall that the ideal is generated by relations (see this entry (http://planetmath.org/PropertiesOfAdmissibleIdeals)) .
Definition. A representation of over with linear mappings is said to be bound by if
for every .
It can be easily checked, that this definition does not depend on the choice of (relation) generators of .
|Title||representations of a bound quiver|
|Date of creation||2013-03-22 19:16:51|
|Last modified on||2013-03-22 19:16:51|
|Last modified by||joking (16130)|