for its definition in the derived category. It is an avatar of the Auslander–Reiten translation in module categories, see [@Malicki] and [@Platzeck] in this volume, and also [@ARS] and [@ASS].

### Dynkin case

We can say even more about the structure of ${\mathcal{D}}^b({\mbox{{\rm mod \!}}}kQ)$ if $Q$ is an orientation of a simply-laced Dynkin diagram: $$\xymatrix@!0{ A_n : & 1\ar@{-}[r] & 2\ar@{-}[r] & \ldots\ar@{-}[r] & n & \\
                  & 1\ar@{-}[dr] & &&& \\
            D_n : & & 3\ar@{-}[r] & 4\ar@{-}[r] & \ldots\ar@{-}[r] & n \\
                  & 2\ar@{-}[ur] \\
                  &&& 4\ar@{-}[d] \\
            E_6 : & 1\ar@{-}[r] & 2\ar@{-}[r] & 3\ar@{-}[r] & 5\ar@{-}[r] & 6 \\
                  &&& 4\ar@{-}[d] \\
            E_7 : & 1\ar@{-}[r] & 2\ar@{-}[r] & 3\ar@{-}[r] & 5\ar@{-}[r] & 6\ar@{-}[r] & 7\\
                  &&& 4\ar@{-}[d] \\
            E_8 : & 1\ar@{-}[r] & 2\ar@{-}[r] & 3\ar@{-}[r] & 5\ar@{-}[r] & 6\ar@{-}[r] & 7\ar@{-}[r] & 8 \\
 }$$

For any quiver $Q$, define the *repetition quiver* ${\mathbb{Z}}Q$ as follows:

-   vertices of ${\mathbb{Z}}Q$ are elements $(i,n)$ of $Q_0\times {\mathbb{Z}}$;

-   for every arrow $a:i\to j$ in $Q$ and every integer $n$, there are arrows $(a,n):(i,n)\to (j,n)$ and $(a^*,n):(j,n)\to (i,n+1)$ in ${\mathbb{Z}}Q$.

If $Q = 1\to 2\to 3\to 4$ is a quiver of type $A_4$, then ${\mathbb{Z}}Q$ 