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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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I can't speak for an original motivation, but from a modern (geometric) point of view, the reason the preprojective algebra is an interesting object is that the moduli stack of representations of the preprojective algebra is equal to the cotangent bundle of the moduli stack of representations of the original quiver Q.

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thanks for the answer, I really appreciated this insight. do you have a reference for this? –  user125763 Apr 26 at 2:53
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