# modules over bound quiver algebra and bound quiver representations

Let $(Q,I)$ be a bound quiver over a fixed field $k$. Denote by $\mathrm{Mod}A$ (resp. $\mathrm{mod}A$) the category^{} of all (resp. all finite-dimensional) (right) modules over algebra^{} $A$ and by ${\mathrm{REP}}_{Q,I}$ (resp. ${\mathrm{rep}}_{Q,I}$) 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

$$F:\mathrm{Mod}A\to {\mathrm{REP}}_{Q,I}$$ |

which restricts to the equivalence of categories

$${F}^{\prime}:\mathrm{mod}A\to {\mathrm{rep}}_{Q,I}.$$ |

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 ${e}_{a}$ be a stationary path in $a\in {Q}_{0}$ and put ${\u03f5}_{a}={e}_{a}+I\in A$. Now if $M$ is a module in $\mathrm{Mod}A$, then define a representation

$$F(M)=({M}_{a},{M}_{\alpha})$$ |

by putting ${M}_{a}=M{\u03f5}_{a}$ ($M$ is a right module over $A$). Now for an arrow $\alpha \in {Q}_{1}$ define ${M}_{\alpha}:{M}_{s(\alpha )}\to {M}_{t(\alpha )}$ by putting ${M}_{\alpha}(x)=x\overline{\alpha}$, where $\overline{\alpha}=\alpha +I\in A$. It can be shown (see [1]) that $F(M)$ is a bound representation.

On module morphisms^{} $F$ acts as follows. If $f:M\to {M}^{\prime}$ is a module morphism, then define

$$F(f)={({f}_{a})}_{a\in {Q}_{0}}$$ |

where ${f}_{a}:{M}_{a}\to {M}_{a}^{\prime}$ is a restriction^{}, i.e. ${f}_{a}(x)=f(x)$. It can be shown that ${f}_{a}$ is well-defined (i.e. ${f}_{a}(x)\in {M}_{a}^{\prime}$) and in this manner $F$ is a functor.

The inverse^{} functor is defined on objects as follows: for a representation $({M}_{a},{M}_{\alpha})$ put

$$G(M)=\underset{a\in {Q}_{0}}{\oplus}{M}_{a}.$$ |

Now we will define right $kQ$-module structure^{} on $G(M)$. For a stationary path ${e}_{a}$ in $a\in {Q}_{0}$ and for $x=({x}_{a})\in G(M)$ put

$$x\cdot {e}_{a}={x}_{a}.$$ |

Now for a path $w=({a}_{1},\mathrm{\dots},{a}_{n})$ from $a$ to $b$ in $kQ$ we consider the evaluation map (see this entry (http://planetmath.org/RepresentationsOfABoundQuiver) for details) ${f}_{w}:{M}_{a}\to {M}_{b}$ and we put

$${(x\cdot w)}_{c}={\delta}_{bc}{f}_{w}({x}_{a}),$$ |

where ${\delta}_{bc}$ 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 $f=({f}_{a}):M\to {M}^{\prime}$ is a morphism of representations then we define

$$G(f)=\underset{a\in {Q}_{0}}{\oplus}{f}_{a}:G(M)\to G(M).$$ |

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]. $\mathrm{\square}$

Corollary. If $Q$ is a finite, connected and acyclic quiver, then there exists an equivalence of categories $\mathrm{Mod}kQ\simeq {\mathrm{REP}}_{Q}$ which restricts to the equivalence of categories $\mathrm{mod}kQ\simeq {\mathrm{rep}}_{Q}$.

Proof. Since $Q$ is finite and acyclic, then the zero ideal^{} $I=0$ is admissible (because lengths of paths are bounded, so ${R}_{Q}^{m}=0$ for some $m\u2a7e1$, where ${R}_{Q}$ denotes the arrow ideal). The thesis follows from the theorem. $\mathrm{\square}$

## 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 |
---|---|

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 |