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 categoryMathworldPlanetmath of all (resp. all finite-dimensional) (right) modules over algebraPlanetmathPlanetmath A and by REPQ,I (resp. repQ,I) the category of all (resp. all finite-dimensional, (see this entry ( 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


which restricts to the equivalence of categories


Sketch of the proof. We will only define functorMathworldPlanetmath F and its quasi-inversePlanetmathPlanetmath 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 aQ0 and put ϵa=ea+IA. Now if M is a module in ModA, then define a representation


by putting Ma=Mϵa (M is a right module over A). Now for an arrow αQ1 define Mα:Ms(α)Mt(α) by putting Mα(x)=xα¯, where α¯=α+IA. It can be shown (see [1]) that F(M) is a bound representation.

On module morphismsMathworldPlanetmath F acts as follows. If f:MM is a module morphism, then define


where fa:MaMa is a restrictionPlanetmathPlanetmath, i.e. fa(x)=f(x). It can be shown that fa is well-defined (i.e. fa(x)Ma) and in this manner F is a functor.

The inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath functor is defined on objects as follows: for a representation (Ma,Mα) put


Now we will define right kQ-module structureMathworldPlanetmath on G(M). For a stationary path ea in aQ0 and for x=(xa)G(M) put


Now for a path w=(a1,,an) from a to b in kQ we consider the evaluation map (see this entry ( for details) fw:MaMb and we put


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):MM is a morphism of representations then we define


It can be shown that G(f) is indeed an A-homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 ModkQREPQ which restricts to the equivalence of categories modkQrepQ.

Proof. Since Q is finite and acyclic, then the zero idealMathworldPlanetmathPlanetmath I=0 is admissible (because lengths of paths are bounded, so RQm=0 for some m1, where RQ denotes the arrow ideal). The thesis follows from the theorem.


Title modules over bound quiver algebra and bound quiver representations
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