## Crystals from KLR and preprojective algebras

Last update: 11 October 2012

## Dynkin diagrams

$G$ a complex reductive algebraic group corresponds to

$(𝔥ℤ,W0) with 𝔥za freeℤ-module W0a finite subgroup of GL(𝔥ℤ) generated by reflections$

Let ${C}_{0}$ be a fundamental region for the action of ${W}_{0}$ on ${𝔥}_{ℝ}=ℝ{\otimes }_{ℤ}{𝔥}_{ℤ}\text{.}$ Let

$𝔥α1,…, 𝔥αn be the walls ofC0$

Make a graph with vertices $1,\dots ,n$ and

$i j i j i j i j if𝔥αi ∡𝔥αjis π2 π3 π4 π6$

Example Type $S{L}_{3}$

$C0 𝔥α1∨ 𝔥α2∨ π3 1 2$

## The quiver Hecke algebra:

Choose an orientation.

$Q= I={colours}= {vertices ofQ}.$
$𝔥ℤ≥0=ℤ≥0 -span{αi∣i∈I}.$

Let $\gamma \in {𝔥}_{ℤ}^{\ge 0}\text{.}$ The quiver Hecke algebra ${R}_{\gamma }$ has generators ${y}_{1},\dots ,{y}_{d},{e}_{u}$ for $u\in {I}^{\gamma },$ ${\psi }_{1},\dots ,{\psi }_{d-1}$ and relations ${y}_{i}{y}_{j}={y}_{j}{y}_{i},$ ${e}_{u}{e}_{v}={\delta }_{uv}{e}_{u},$ $1=\sum _{u\in {I}^{\gamma }}{e}_{u}$ and more... where

$Iγ= { u=(u1,…,ud) sequences of colors withαu1+…+ αud=γ }$

with $ℤ$-grading

$deg(eu), deg(yi)=2, deg(ψreu)= { -2, ifur= ur+1 -1, if ur ur+1 0, if ur ur+1$

Let $R=\underset{\gamma \in {ℤ}_{\ge 0}}{⨁}{R}_{\gamma }$

$MaℤgradedR -module so thatM=⨁j∈ℤ M[j]$

Define

$char(M)= ∑j∈ℤ ∑u∈Iγ dim(euM[J]) qjfu1… fud.$

(generating function in noncommutative ${f}_{i},\phantom{\rule{0.2em}{0ex}}i\in I\text{).}$

(Khoranov-Lavda/Rouquier)

$char: Grothendieck group { fin. dim'lℤ-graded Rmodules } ⟶ Uqn- (quantum group) simple ℝ-modules Lb ⟼ char(Lb) (canonical basis)$

Define ${\stackrel{\sim }{f}}_{i}b$ by

$Lf∼ib= head ( Ind Rαi⊗Rγ Rαi+γ (Lb) )$

As directed graphs with labels $\stackrel{{\stackrel{\sim }{f}}_{i}}{⟶}$

${ simpleℤ-graded R-modulesLb } ⟶∼ { MV polytopes b } .$

## Preprojective algebras

$Q= Q‾=$

Idea: Replace beads by vector spaces.

$ℂn1 ℂn2 ℂn3 ℂnjcorresponds to njbeads on runnerj.$

The data of

1. a vector space for each vertex
2. a linear transformation for each edge

is a representation of $Q$ (or $\stackrel{‾}{Q}\text{).}$

In the case of $\stackrel{‾}{Q}$ require

$∑ i a ∈Q a*a= ∑ i a ∈Q aa*,for eachi∈I.$

Example: Type ${GL}_{n}$

$Q= a1 a2 a3 an-2 Q‾= a1 a2 an-2 a1* a2* an-2*$

and we require

$a1a1*=0, ai-1*ai-1= aiai*for i=2,…,n-2, an-2*an-2=0$

Let

$Λ= {isomorphism classes of representations ofQ‾satisfying(PP)}$

As discrete $d$ graphs with labels $\stackrel{\sim }{{f}_{i}},$

${ irreducible components ΛbofΛ } ⟶∼ { MV polytopes b }$

## Notes and References

These are a typed version of lecture notes for the fifth in a series of lectures given at the Brazil Algebra Conference, Salvador, 19 July 2012.