modules over bound quiver algebra and bound quiver representations
Let (Q,I) be a bound quiver over a fixed field k. Denote by ModA (resp. modA) the category of all (resp. all finite-dimensional) (right) modules over algebra
A and by REPQ,I (resp. repQ,I) the category of all (resp. all finite-dimensional, (see this entry (http://planetmath.org/QuiverRepresentationsAndRepresentationMorphisms) for details) bound representations.
We will also allow I=0 (which is an admissible ideal only if lengths of paths in Q are bounded, in particular when Q is finite and acyclic). In this case bound representations are simply representations.
Theorem. If Q is a connected and finite quiver, I and admissible ideal in kQ and A=kQ/I, then there exists a k-equivalence of categories
F:ModA→REPQ,I |
which restricts to the equivalence of categories
F′:modA→repQ,I. |
Sketch of the proof. We will only define functor F and its quasi-inverse
G. For proof that F is actually an equivalence please see [1, Theorem 1.6] (this not difficult, but rather technical proof).
Let ea be a stationary path in a∈Q0 and put ϵa=ea+I∈A. Now if M is a module in ModA, then define a representation
F(M)=(Ma,Mα) |
by putting Ma=Mϵa (M is a right module over A). Now for an arrow α∈Q1 define Mα:Ms(α)→Mt(α) by putting Mα(x)=xˉα, where ˉα=α+I∈A. It can be shown (see [1]) that F(M) is a bound representation.
On module morphisms F acts as follows. If f:M→M′ is a module morphism, then define
F(f)=(fa)a∈Q0 |
where fa:Ma→M′a is a restriction, i.e. fa(x)=f(x). It can be shown that fa is well-defined (i.e. fa(x)∈M′a) and in this manner F is a functor.
The inverse functor is defined on objects as follows: for a representation (Ma,Mα) put
G(M)=⊕a∈Q0Ma. |
Now we will define right kQ-module structure on G(M). For a stationary path ea in a∈Q0 and for x=(xa)∈G(M) put
x⋅ea=xa. |
Now for a path w=(a1,…,an) from a to b in kQ we consider the evaluation map (see this entry (http://planetmath.org/RepresentationsOfABoundQuiver) for details) fw:Ma→Mb and we put
(x⋅w)c=δbcfw(xa), |
where δbc denotes the Kronecker delta. It can be shown that G(M) is a kQ-module with the property that G(M)I=0. In particular G(M) is a kQ/I-module.
Now, if f=(fa):M→M′ is a morphism of representations then we define
G(f)=⊕a∈Q0fa:G(M)→G(M). |
It can be shown that G(f) is indeed an A-homomorphism and that G is a functor.
Also, it follows easily from definitions that both F and G 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 Q is a finite, connected and acyclic quiver, then there exists an equivalence of categories ModkQ≃REPQ which restricts to the equivalence of categories modkQ≃repQ.
Proof. Since Q is finite and acyclic, then the zero ideal I=0 is admissible (because lengths of paths are bounded, so RmQ=0 for some m⩾, where denotes the arrow ideal). The thesis follows from the theorem.
References
- 1 I. Assem, D. Simson, A. SkowroÃÆski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007
Title | modules over bound quiver algebra and bound quiver representations |
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Canonical name | ModulesOverBoundQuiverAlgebraAndBoundQuiverRepresentations |
Date of creation | 2013-03-22 19:17:34 |
Last modified on | 2013-03-22 19:17:34 |
Owner | joking (16130) |
Last modified by | joking (16130) |
Numerical id | 5 |
Author | joking (16130) |
Entry type | Theorem |
Classification | msc 14L24 |