\Sigma X$.

On the right of the picture, we see a “hook” of solid arrows, which induces a triangle $U\to F\to V\to \Sigma U$. The rule that “hooks” must obey is the following: the length of the second part of the hook (from $F$ to $V$ on the picture) is one more than the length of the downward path from the first object (here $U$) to the bottom of the picture. Of course, hooks that are symmetric to the one pictured also yield triangles.

Cluster categories {#cluster-categories}
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Cluster categories are triangulated categories that share many of the combinatorial properties of cluster algebras. They constitute the main setting for the definition of cluster characters (see Section \[sect::cluster-characters\]).

### Orbit categories

Let ${\mathcal{C}}$ be a $k$-linear category, and let $F$ be an automorphism of ${\mathcal{C}}$. The *orbit category* ${\mathcal{C}}/F$ is the $k$-linear category defined as follows:

-   its objects are the objects of ${\mathcal{C}}$;

-   for any objects $X$ and $Y$, ${\operatorname{{\rm Hom}}_{{\mathcal{C}}/F}}(X,Y):= \bigoplus_{n\in {\mathbb{Z}}}{\operatorname{{\rm Hom}}_{{\mathcal{C}}}}(X, F^n Y)$.

As one might expect from the name “orbit category”, the objects $X$ and $FX$ become isomorphic in ${\mathcal{C}}/F$.

### Cluster categories {#cluster-categories-1}

Let $Q$ be a quiver without oriented cycles. The *cluster category 