Canonical bases and Khovanov-Lauda algebras

In [KL1], [KL2] Khovanov and Lauda have introduced a new family of algebras and formulated some conjecture predicting a connection between the representation theory of these algebras and Lusztig’s geometric construction of canonical bases. The goal of this paper is to prove part of this conjecture.

Before to go on let us make a few historical remarks. The KLR-algebras were first introduced by Khovanov and Lauda in [KL1], [KL2] with some restrictions on the quiver. There were independently discovered by Rouquier, in a more general version. See Remark 3.3 below for details and [R] for the definition and the first properties. After our paper was written Rouquier informed us he has also obtained the same result as ours, independently.

We use the following notation for $q$-numbers

If no confusion is possible we’ll abbreviate

Reminder on quivers, extensions and convolution algebras

We give a few notation on quivers and equivariant homology. Given a quiver $\Gamma$ we first recall the definition of the semisimple complexes on the moduli stack of representations of $\Gamma$ introduced by Lusztig in [L1]. Next we define their Ext-algebras (with respect to the Yoneda product). Finally we recall a lemma of Ginzburg which relates Ext-algebras to convolution algebras in equivariant homology.

1. Representations of quivers

Let $\Gamma$ be a finite nonempty quiver such that no arrow may join a vertex to itself. Recall that $\Gamma$ is a tuple $(I,H,h\mapsto h^{\prime},h\mapsto h^{\prime\prime})$ where $I$ is the set of vertices, $H$ is the set of arrows and for each $h\in H$ the vertices $h^{\prime},h^{\prime\prime}\in I$ are the origin and the goal of $h$ respectively. Although this hypothesis is not necessary, we’ll always assume that the set $I$ is finite in the rest of the paper. For each $i,j\in I$ we write

We’ll abbreviate $i\to j$ for $H_{i,j}\neq\emptyset$, $i\not\to j$ for $H_{i,j}=\emptyset$, and $h:i\to j$ for $h\in H_{i,j}$. Let $h_{i,j}$ be the number of elements in $H_{i,j}$ and set

identifies $Y_{\nu}$ with a set of sequences

2. Constructible sheaves

There is a lot of literature on the category ${\boldsymbol{{\Cal{D}}}}_{G}(X)$, see [BL], [J], [L3, sec.~1], [L4, sec.~1] for instance. Although we’ll only use standards properties of ${\boldsymbol{{\Cal{D}}}}_{G}(X)$ let us recall a few definitions for the comfort of the reader. We’ll use the notation of [BBD] for sheaves and Deligne’s definition of ${\boldsymbol{{\Cal{D}}}}_{G}(X)$, see [BL, sec.~I.2, app.~B], [J, sec.~1.2.3]. More precisely, let $[G\setminus X]$ be the usual simplicial topological set whose $n$-skeleton is $G^{n}\times X$ for each $n$. We put the transcendental topology on $X$, $G$. Let ${\boldsymbol{{\Cal{S}}h}}_{G}(X)$ be the full subcategory of the category of simplicial sheaves on $[G\setminus X]$ for which all structure morphisms are isomorphisms. It is an Abelian category which is equivalent to the category of $G$-equivariant sheaves on $X$ by [D, (6.1.2)]. We define ${\boldsymbol{{\Cal{D}}}}_{G}(X)$ as the full subcategory of the bounded derived category of simplicial sheaves on $[X\setminus G]$ consisting of the complexes whose cohomology sheaves belong to ${\boldsymbol{{\Cal{S}}h}}_{G}(X)$.

Let us recall a few simple facts. If $H\subset G$ is a closed subgroup then the pull back by the obvious morphism of simplicial topological spaces $[H\setminus X]\to[G\setminus X]$ is the forgetful functor ${\boldsymbol{{\Cal{D}}}}_{G}(X)\to{\boldsymbol{{\Cal{D}}}}_{H}(X)$. An object of ${\boldsymbol{{\Cal{D}}}}_{G}(X)$ is perverse if the corresponding object in ${\boldsymbol{{\Cal{D}}}}(X)$ is perverse.

We define the space of $G$-equivariant homology by

Fix a morphism of quasi-projective algebraic $G$-varieties $f:X\to Y$. If $f$ is a proper map there is a direct image homomorphism

If $f$ is a smooth map of relative dimension $d$ there is an inverse image homomorphism

If $X$ has pure dimension $d$ there is a natural homomorphism

3. Ext-algebras

5. Remark

7. Convolution algebras

Recall the following isomorphism, see (1.4)

Proof : Part $(b)$ is proved as in [CG, Prop.~8.6.35]. Note that in loc. cit. both claims are proved for non-equivariant homology. However the proof uses only standard tools and generalizes to the equivariant setting. Part $(a)$ is standard. A proof of $(a)$ is given in Remark 2.21 below for the comfort of the reader.

9. Shift of the grading

10. Notation

1. Notation

3. Identification of 𝔖𝔖\mathfrak{S} with the symmetric group

4. The root system

Let $s_{l}\in\Pi$ denote the reflection with respect to the simple root $\chi_{l}-\chi_{l+1}$. Under the identification $\mathfrak{S}=\mathfrak{S}_{m}$ the reflection $s_{l}$ is taken to the simple transposition $(l,l+1)$.

(c) We have $\bar{O}^{s}_{w,w^{\prime}}=\emptyset$ unless $w^{\prime}=w$ or $ws$.

7. Example

11. Notation

12

13. Reduction to the torus

We’ll write $w(f)$ for the right hand side. This isomorphism is not canonical, because it depends on the choice of $w$.

16. Examples

$(b)$ Let $s_{l}$ be as above. Given $w\in\mathfrak{S}$ we write

We’ll also abbreviate $\psi_{w}$, $\psi_{w,w^{\prime}}$ for the elements

The following lemma is standard. Its proof is left to the reader.

19

Note that $\Lambda^{s}_{w,w^{\prime}}=0$ unless $w^{\prime}=w$ or $ws$ by Lemma 2.6$(c)$. Let $\star$ denote also the convolution products

(a) The forgetful maps below commute with $\star$

(b) For $w,w^{\prime},w^{\prime\prime}\in\mathfrak{S}$ we have

21. Remark

Under this inclusion the map (2.10) is given by

In the rest of the proof we’ll assume that the following hold

Further there is no arrow in $H$ joining $i_{l}$ and $i_{l+1}$. Thus by (2.7) we have

where $g=(f-s(f))/(\chi_{l}-\chi_{l+1}).$

Therefore we get the following alternatives

(note that our notation for flags is opposite to the one in loc. cit.). Thus we have

28. Examples

KLR-algebras

In this section we compute the Ext-algebras introduced in the first section.

3. Remarks

Here we have set $s_{l,l+2}=s_{l}s_{l+1}s_{l}$ if $l\neq m-1,m$ and

Next, fix an integer $l=1,2,\dots m-1$. We claim that

See the definition of $\sigma(l)$ in Section 2.22 for the second identity. To complete the proof of Step 2 we are reduced to prove the following.

On the other hand, since $\Lambda^{w}_{x,xsw}=0$ and

(a) For each $x,y\in\mathfrak{S}$ we have

Proof : Part $(a)$ is left to the reader. Let us prove $(b)$. Set

This equality is a consequence of the following well-known formula

This implies that $A=B$. The second claim follows.

9

In this section, using the computations of Section 3 we prove a conjecture of [KL1] yielding a categorification of the canonical basis of the negative part of Drinfeld-Jimbo quantized enveloping algebra.

Let $1_{\nu,\nu^{\prime}}$ be the image of the identity element by the map (4.1). The induction yields an additive functor

It takes projectives to projectives and it commutes with the shift of the grading. Thus it yields an ${\Cal{A}}$-linear homomorphism

3

By [KL1, prop.~3.4, sec.~3.2] there is an unique ${\Cal{A}}$-algebra isomorphism

The following is the second main result of the paper.

6. Definition of the canonical basis ℬℬ{\Cal{B}}

There is also an unique ${\Cal{A}}$-algebra isomorphism

7

By base change the ${\Cal{A}}$-linear maps ${\gamma}_{\Cal{A}}$, $\mu_{\Cal{A}}$ yield ${\Cal{K}}$-linear maps

The proof of the following lemma is postponed in Section 4.10.

9. Remarks

$(b)$ By (4.3), (4.6) and the previous remark we have

10

Proof of Lemma 4.8 : To prove the first claim we must check that the forgetful map yields an isomorphism

The second claim follows from the first one. Indeed, the forgetful map