TITLE: Motivation for the Preprojective Algebra
QUESTION [18 upvotes]: Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of $\bar{Q}$ be the arrows $Q_1$ together with arrows $\alpha ^* :j\rightarrow i$ for each arrow $\alpha :i\rightarrow j$ in $Q_1$, so $\bar{Q}_1=Q_1\cup Q_1^*$. The $\textbf{preprojective algebra}$ of the quiver $Q$ is defined as 
\begin{equation*} 
\mathcal{P}_k(Q)=k\bar{Q}/(\rho )\nonumber
\end{equation*}
where 
\begin{align*}
\rho = \displaystyle \sum_{\alpha \in Q_1}[\alpha,\alpha^*]=\sum_{\alpha \in Q_1}(\alpha \alpha^*-\alpha^*\alpha)
\end{align*}
Let $\theta = \text{Ext}_{kQ}^1(D(kQ_{kQ}),kQ)$. We then have the isomorphism
\begin{equation*}
\mathcal{P}_k(Q) \cong T_{kQ}(\theta )\nonumber 
\end{equation*}
which acts as identity on $kQ$, and which maps the arrows in $Q_1^*$ onto the augmentation ideal of $T_{kQ}(\theta )$. This is useful since the right side is easier to generalize. It also implies that the preprojective algebra is the sum of the preprojective modules. For a proof of the isomorphism see http://pages.uoregon.edu/njp/Ringel98.pdf or theorem 3.1 in http://www1.maths.leeds.ac.uk/~pmtwc/preproj2.pdf
I am wondering why one studies the preprojective algebra. What was the original motivation? In the paper by Ringel he mentions that they occur naturally in diverse situations, and I was wondering if anyone is familiar with any such examples. The paper by Ringel is quite old so there are probably more recent examples as well.

REPLY [2 votes]: The following expands a little on Hugh's third point.
If $G$ is a finite subgroup of $\mathrm{SL}_2(\mathbb{K})$, where $\mathbb{K}$ is an algebraically closed field of characteristic $0$, then $G$ acts naturally on $\mathbb{K}[x,y]$. We can form the skew group ring $\mathbb{K}[x,y]*G$, which has elements $(f,g)$ for $f\in \mathbb{K}[x,y]$ and $g\in G$, with multiplication
$$(f_1,g_1)(f_2,g_2)=(f_1g(f_2),g_1g_2).$$
Then $\mathbb{K}[x,y]*G$ is Morita equivalent to the preprojective algebra of the McKay graph of $G$, which is an affine Dynkin diagram. (This is the way in which such diagrams classify the finite subgroups of $\mathrm{SL}_2(\mathbb{K})$.) If $G$ is cyclic, corresponding to affine type A, then there is even an isomorphism. This is proved by Reiten and van den Bergh in Tame and maximal orders of finite representation type. (The proof may be a little implicit—unfortunately I don't have access to the paper right now, but Crawley-Boevey and Holland prove a deformed version and credit Reiten–van den Bergh with the classical one. They reference the proof of Prop. 2.13, and say that the 'key calculation' is Prop. 2.4.)
If one works with the complete preprojective algebra instead, then the Morita equivalence is with $S*G$, for $S=\mathbb{K}[[x,y]]$ the power series ring. Letting $R$ be the invariant subring (which is, by definition, a Kleinian singularity), $S$ is a Cohen–Macaulay module over $R$, and in fact (as shown by Herzog) every indecomposable Cohen–Macaulay $R$-module is a summand of $S$ (with some multiplicity). A result of Auslander is that there is an isomorphism
$$S*G\cong \operatorname{End}_R(S),$$
so this endomorphism ring is also Morita equivalent to the appropriate preprojective algebra. (The above endomorphism algebra has global dimension $2$—for this and other reasons, $S$ is a prototypical example of what is now known as a non-commutative crepant resolution.)
A good reference for the above (sadly excluding the part about the preprojective algebra!), including more general statements and some more geometry, is Leuschke and Wiegand's book on Cohen–Macaulay representations, particularly chapters 5 and 6.