# 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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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? – bananastack Apr 26 '14 at 2:53

Here is one of the reasons (maybe the only reason) for the name. The preprojective algebra of the path algebra $H=K\mathcal Q$ can also be defined as follows as a graded algebra: $P(\mathcal Q)=\bigoplus_{n\ge 0}\operatorname{Hom}_H(H,\tau^{-n}H)$ where $\tau$ is the Auslander-Reiten translate.

As an $H$-module it is isomorphic to the direct sum of all the preprojective $H$-modules. The multiplication for this definition is given by the following:

if $\mu\colon H\to\tau^{-i}H$ and $\eta\colon H\to\tau^{-j}H$, then we take $\eta\mu$ to be the composition $(\tau^{-i}\eta)\mu$.

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Preprojective algebras are a key ingredient in Lusztig's semicanonical basis for U(n). There is a short explanation here: http://arxiv.org/pdf/1009.4552.pdf, and references, but I am not sure what the best place to look is.

Preprojective algebras have been very successfully used to categorify cluster algebras (see work of Geiss-Leclerc-Schröer, eg the survey: http://arxiv.org/abs/0804.3168)

Affine-type preprojective algebras come up naturally in the McKay correspondence. I don't have a good reference for this.

It is sometimes useful to be able to wrap up (nice) subcategories of representations of quivers into single (indecomposable) modules over the preprojective algebra. The example I am thinking of is a paper I wrote with Steffen Oppermann and Idun Reiten, which originated out of a desire to prove a conjecture about quiver representations (http://arxiv.org/abs/1205.3268).

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