Preprojective algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 17 February 2011

Representations of quivers

Let $Q$ be a Dynkin diagram of type $A,D$ or $E$ with vertex set $I$. For $\gamma ={\gamma }_1{\alpha }_1^{\vee }+\cdots +{\gamma }_n{\alpha }_n^{\vee }$, with ${\gamma }_1,\dots ,{\gamma }_n\in {\mathbb{Z}}_{\ge 0}$, define

$Q_{\gamma }=\{M\in Q\mathrm{-mod}|\dim \left(M\right)=\gamma \}$     and     $G_{\gamma }=\prod _{i=1}^n{\mathrm{GL}}_{{\gamma }_i}\left(ℂ\right)$.

Preprojective algebras

Let $Q$ be a Dynkin diagram of type $A,D$ or $E$ with vertex set $I$. Let $\bar{Q}$ be the quiver with an extra edge $a^{*}$, in the opposite direction, for each edge $a$ in $Q$. Let $ℂ\bar{Q}$ be the path algebra of $\bar{Q}$. For each vertex $i$ of $Q$ let

$r_i={\displaystyle \sum _{a\in Q,s\left(a\right)=i}{a}^{*}a-\sum _{a\in Q,e\left(a\right)=i}{a}^{*}a},$

(pprels)

where $s\left(a\right)$ is the source of $a$ and $e\left(a\right)$ is the target of $a$. The preprojective algebra is

${\displaystyle \Lambda =\frac{ℂ\bar{Q}}{⟨r_i|i\in I⟩}}$.

(ppalg)

For $\gamma ={\gamma }_1{\alpha }_1^{\vee }+\cdots +{\gamma }_n{\alpha }_n^{\vee }$, with ${\gamma }_1,\dots ,{\gamma }_n\in {\mathbb{Z}}_{\ge 0}$, define

${\Lambda }_{\gamma }=\{M\in \Lambda \mathrm{-mod}|\dim \left(M\right)=\gamma \}$     and     $G_{\gamma }=\prod _{i=1}^n{\mathrm{GL}}_{{\gamma }_i}\left(ℂ\right)$.

The preprojective cycles of type $\gamma$ are the irreducible components

${\Lambda }_b\in \mathrm{Irr}\left({\Lambda }_{\gamma }\right)$.

(ppcyc)

The character of a $\Lambda$-module $M$ is the element of the shuffle algebra given by

$\mathrm{ch}\left(M\right)={\displaystyle \sum _{i_1,\dots i_d}\chi \left({ℱ\left(M\right)}_{i_1,\dots ,i_d}\right)f_{i_1}\cdots f_{i_{\mathrm{d}}}}$,

(Λch)

where ${ℱ\left(M\right)}_{i_1,\dots ,i_d}$ is the subset of the full flag variety of $M$ of composition series of $M$ of type $\left(i_1,\dots ,i_d\right)$ and $\chi \left({ℱ\left(M\right)}_{i_1,\dots ,i_d}\right)$ is the Euler characteristic of ${ℱ\left(M\right)}_{i_1,\dots ,i_d}$.

Let

$\stackrel{\sim }{ℳ}=\underset{\gamma }{\oplus }{M\left({\Lambda }_{\gamma }\right)}^{G_{\gamma }}$,      where

$M\left({\Lambda }_{\gamma }\right)$ is the space of constructible functions on ${\Lambda }_{\gamma }$ and ${M\left({\Lambda }_{\gamma }\right)}^{G_{\gamma }}$ is the subspace of constructible functions constant on the orbits of $G_{\gamma }$. Let

$f_i=\mathrm{ch}\left(Z_{{\alpha }_i^{\vee }}\right)$,      where    $Z_{{\alpha }_i^{\vee }}={\Lambda }_{{\alpha }_i^{\vee }}\cong \mathrm{pt}$,

and let $ℳ$ be the subalgebra of $\stackrel{\sim }{ℳ}$ generated by the $f_i$. By [Lu2, Theorem 12.13] the map

$\begin{array}{ccc}U{𝔫}^{-}& \to & ℳ\\ f_i& \mapsto & f_i\end{array}$      is an algebra isomorphism.

The $a^{*}$-forgetting morphism is

${\pi }_{\gamma }:{\Lambda }_{\gamma }\to Q_{\gamma }$

According to [GLS, §10.3], it was proved by Lusztig [Lu1] that there are bijections

$\begin{array}{ccccc}\left\{{\Lambda }_b\right\}& \leftrightarrow & \left\{G_{\gamma }\text{-orbits on}{Q}_{\gamma }\right\}& \leftrightarrow & \{\text{multisegments}bof\; weight\gamma \}\\ \overline{{\pi }^{-1}\left(Q_b\right)}& \leftarrow & Q_b& \leftarrow & b\end{array}$.

The type $A_n$ quiver

Let $Q_n$ be

$•\stackrel{\phantom{-}{a}_{1\phantom{-}}}{\leftarrow }•\stackrel{\phantom{-}{a}_{2\phantom{-}}}{\leftarrow }•\stackrel{\phantom{-}{a}_{3\phantom{-}}}{\leftarrow }\cdots \stackrel{a_{n-2}}{\leftarrow }•\stackrel{a_{n-1}}{\leftarrow }•$

Then ${\bar{Q}}_n$ is

$•\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{\leftrightarrows }}•\underset{\phantom{-}{a}_{2\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{2\phantom{-}}}{\leftrightarrows }}•\underset{\phantom{-}{a}_{3\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{3\phantom{-}}}{\leftrightarrows }}\cdots \underset{a_{n-2}^{*}}{\overset{a_{n-2}}{\leftrightarrows }}•\underset{a_{n-1}^{*}}{\overset{a_{n-1}}{\leftrightarrows }}•$

and the preprojective algebra $\Lambda$ is generated by $a_1,\dots ,a_n,a_1^{*},\dots ,a_n^{*},$ with relations

$a_1{a}_1^{*}=0$, $a_{n-1}^{*}{a}_{n-1}=0$,     and     $a_i^{*}{a}_i=a_{i+1}{a}_{i+1}^{*}$.

A typical $\Lambda$-module can be viewed as a linear combination of the basis elements with "red" maps corresponding to the $a_i$ and "blue" maps corresponding to the $a_i^{*}$. If we view the basis elements as boxes, then the $a^{*}$-forgetting morphism takes

    to    

Example: Type $A_2$. In this case

$Q_2\text{is}•\stackrel{\phantom{-}{a}_{1\phantom{-}}}{\leftarrow }•\text{and}{\bar{Q}}_2\text{is}•\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{\leftrightarrows }}•$.

Then

${\Lambda }_{{\alpha }_1}=\left\{\stackrel{ℂ}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{\leftrightarrows }}\stackrel{0}{•}\right\}\simeq \mathrm{pt}$         and         ${\Lambda }_{{\alpha }_2}=\left\{\stackrel{0}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{\leftrightarrows }}\stackrel{ℂ}{•}\right\}\simeq \mathrm{pt}$

and

${\Lambda }_{{\alpha }_1+{\alpha }_2}=\{\stackrel{ℂ}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{\leftrightarrows }}\stackrel{ℂ}{•}|a_1,a_1^{*}\in ℂ\text{with}{a}_1{a}_1^{*}=0\}$

has two irreducible components

${\Lambda }_{(1,0,1)}=\left\{\right(a,0)|a\in ℂ\}$ and ${\Lambda }_{(0,1,0)}=\left\{\right(0,a^{*})|a^{*}\in ℂ\}$

and $G_{{\alpha }_1+{\alpha }_2}={ℂ}^{\times }\times {ℂ}^{\times }$ acts on ${\Lambda }_{{\alpha }_1+{\alpha }_2}$ with three orbits

${\Lambda }_{(1,0,1)}-\left\{\right(0,0\left)\right\},{\Lambda }_{(0,1,0)}-\left\{\right(0,0\left)\right\},\text{and}\left\{\right(0,0\left)\right\}$.

and, when $\gamma =2{\alpha }_1+2{\alpha }_2$,

${\Lambda }_{2{\alpha }_1+2{\alpha }_2}=\{\stackrel{{ℂ}^2}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{\leftrightarrows }}\stackrel{{ℂ}^2}{•}|a_1,a_1^{*}\in ℂ\text{with}{a}_1{a}_1^{*}=0\}$

has three irreducible components

${\Lambda }_{(1,1,1)}=\left\{\right(a,a^{*})\in {\Lambda }_{\gamma }|\mathrm{rk}\left(a\right)\le 1\text{and}\mathrm{rk}\left(a^{*}\right)\le 1\},$

$=$

${\Lambda }_{(2,0,2)}=\left\{\right(a,a^{*})\in {\Lambda }_{\gamma }|a^{*}=0\}$    and    ${\Lambda }_{(0,2,0)}=\left\{\right(a,a^{*})\in {\Lambda }_{\gamma }|a=0\}$

As computed in [GLS, §5.6],

$\mathrm{ch}\left({\Lambda }_{(1,0,1)}\right)=f_1\circ f_2\text{and}\mathrm{ch}\left({\Lambda }_{(0,1,0)}\right)=f_2\circ f_1,$

and

$\mathrm{ch}\left({\Lambda }_{(2,0,2)}\right)=\frac{1}{4}{f}_1\circ f_1\circ f_2\circ f_2,\mathrm{ch}\left({\Lambda }_{(0,2,0)}\right)=\frac{1}{4}{f}_2\circ f_2\circ f_1\circ f_1,$

and

$\mathrm{ch}\left({\Lambda }_{(1,1,1)}\right)=\frac{1}{2}{f}_2\circ f_1\circ f_1\circ f_2,=\frac{1}{2}{f}_1\circ f_2\circ f_2\circ f_1,$

Notes and References

This summary of the theory of quiver representations and preprojective algebras is part of joint work with A. Ghitza and S. Kannan on the relationship between MV-cycles and the Borel-Weil-Bott theorem. The theory began from the ideas of [Lu1-2] and has been developed at length in [GLS] and later papers of Geiss, Leclerc and Schroer.

References

[GLS] C. Geiss, B. Leclerc and J. Schröer, Semicanonical bases and preprojective algebras, Ann. Sc. École Norm. Sup. 38 (2005), 193-253. (2003), 567-588, arXiv:math/0402448, MR2144987.

[Lu1] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365-421.

[Lu2] G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), 129-139.

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