modules over bound quiver algebra and bound quiver representations
Let be a bound quiver over a fixed field . Denote by (resp. ) the category of all (resp. all finite-dimensional) (right) modules over algebra and by (resp. ) the category of all (resp. all finite-dimensional, (see this entry (http://planetmath.org/QuiverRepresentationsAndRepresentationMorphisms) for details) bound representations.
We will also allow (which is an admissible ideal only if lengths of paths in are bounded, in particular when is finite and acyclic). In this case bound representations are simply representations.
which restricts to the equivalence of categories
Let be a stationary path in and put . Now if is a module in , then define a representation
On module morphisms acts as follows. If is a module morphism, then define
where is a restriction, i.e. . It can be shown that is well-defined (i.e. ) and in this manner is a functor.
The inverse functor is defined on objects as follows: for a representation put
Now we will define right -module structure on . For a stationary path in and for put
Now for a path from to in we consider the evaluation map (see this entry (http://planetmath.org/RepresentationsOfABoundQuiver) for details) and we put
where denotes the Kronecker delta. It can be shown that is a -module with the property that . In particular is a -module.
Now, if is a morphism of representations then we define
It can be shown that is indeed an -homomorphism and that is a functor.
Also, it follows easily from definitions that both and take finite-dimensional objects to finite-dimensional.
It remains to show that these two functors are quasi-inverse. For the proof please see [1, Theorem 1.6].
Corollary. If is a finite, connected and acyclic quiver, then there exists an equivalence of categories which restricts to the equivalence of categories .
Proof. Since is finite and acyclic, then the zero ideal is admissible (because lengths of paths are bounded, so for some , where denotes the arrow ideal). The thesis follows from the theorem.
|Title||modules over bound quiver algebra and bound quiver representations|
|Date of creation||2013-03-22 19:17:34|
|Last modified on||2013-03-22 19:17:34|
|Last modified by||joking (16130)|