Motivation for the Preprojective Algebra

Question

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.

Answer 1

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.

Answer 2

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 $(\tau^{-i}\eta)\mu$.

Answer 3

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).

Answer 4

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.

Answer 5

There is a categorical enhancement of the current accepted answer:

There is a notion of $n$-Calabi-Yau completion $\Pi_n(\mathcal{C})$ of any dg/abelian category $\mathcal{C}$. It's $n$-Calabi-Yau and its derived moduli stack of objects is $$\mathcal{M}_{\Pi_n\mathcal{C}}\ =\ T^*[-n+2]\mathcal{M}_{\mathcal{C}}$$ i.e. the shifted cotangent complex of the moduli stack of $\mathcal{C}$.

Example. We have that $$\Pi_n(\text{Coh}(X))\ =\ \text{Coh}(K_X)$$ whenever $X$ is a say smooth variety of dimension $n-1$. So if $C$ is a smooth curve, $\Pi_2(\text{Coh}(C))= \text{Coh}(T^*C)$.

Example. We have another one dimensional category, $\text{Rep}Q$. Moreover, $$\Pi_2(\text{Rep}Q)\ =\ \text{Rep}\Pi_Q,$$ where $\Pi_Q$ is the preprojective algebra. Thus, on moduli stacks, we have that $\mathcal{M}_{\text{Rep}\Pi_Q}$ is the cotangent complex $T^*\mathcal{M}_{\text{Rep} Q}$.