# Motivation for the Preprojective Algebra

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.

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