Data for quiver Hecke algebras

## Data for quiver Hecke algebras

1. $ℱ$ is the free algebra generated by ${f}_{i},i\in I.$ $Q + = ∑ i∈ I ℤ ≥0 αi with deg f i 1 … f i d = α i 1 + … + α i d .$ The symmetric group ${S}_{d}=⟨{\sigma }_{1},\dots ,{\sigma }_{d-1}⟩$ acts on (by rearrangements) with orbit decomposition $I d = ⊔ α∈ Q + ,ht α =d I α where I α = u∈ I* | deg u =α .$
2. Fix a symmetric bilinear form $⟨,⟩:{Q}^{+}×{Q}^{+}\to ℤ$ given by values $⟨{\alpha }_{i},{\alpha }_{j}⟩\in ℤ$ so that $A= αi ∨ αj with αi ∨ = 2 αi αi αi$ is a Cartan matrix for a symmetrisable Kac-Moody Lie algebra.
3. $\Gamma$ is the graph with vertices $I$ and edges $i$ $j$ if $⟨{\alpha }_{i},{\alpha }_{j}⟩\ne 0.$ Fix an orientation $ε ij$ where ${\epsilon }_{ij}=+1$ if $i$ $j$ and ${\epsilon }_{ij}=-1$ if $i$ $j$ and set

## Quiver Hecke algebras ${ℛ}_{\alpha },\alpha \in {Q}^{+}$

${ℛ}_{\alpha }$ is the associative $ℤ$-graded algebra given by generators $e u , x 1 e u ,…, x d e u , τ 1 e u ,…, τ d-1 e u ,u ∈ I α$ with degrees $deg e u =0,deg x i e u = u i u i ,deg τi eu =- u i u i +1$ where ${u}_{i}$ is the $i$th letter in $u,$ and relations $e u e v = δ uv , ∑ u∈ I α e u =1, x i x j = x j x i , x i e u = e u x i ,$$τ i e u = e σi u τ i , τ i τ j = τ j τ i ,ifi≠i,i±1$$τ i 2 e u = Q u i , u i+1 x i x i+1 e u ,$

Note: ${x}_{i}=\sum _{u\in {I}^{\alpha }}{x}_{i}{e}_{u},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\tau }_{i}=\sum _{u\in {I}^{\alpha }}{\tau }_{i}{e}_{u}.$

## Structure of ${ℛ}_{\alpha }$

For each $\sigma \in {S}_{\alpha }$ fix a reduced word $\sigma ={\sigma }_{{i}_{1}}\dots {\sigma }_{{i}_{l}}$ and set ${\tau }_{\sigma }={\tau }_{{i}_{1}}\dots {\tau }_{{i}_{l}}.$

(Khovanov-Lauda Rouquier)
${ℛ}_{\alpha }$ has basis $\left\{{x}_{1}^{{n}_{1}}\dots {x}_{d}^{{n}_{d}}{\tau }_{\sigma }{e}_{u}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}u\in {I}^{\alpha },\sigma \in {S}_{d},{n}_{1},\dots ,{n}_{d}\in {ℤ}_{\ge 0}\right\}.$ If $\alpha \in {Q}^{+}$ and $\beta \in {Q}^{+}$ then $I α+β = ∪ σ∈ S k+l S k × S l σ I α . I β$ and there is a homomorphism $ℛ α ⊗ ℛ β → ℛ α+β eu ⊗ e v ↦ x i e uv eu ⊗ e v ↦ x i e uv eu ⊗ x j e v ↦ x j+k e uv τ i eu ⊗ e v ↦ τ i e uv eu ⊗ τ j e v ↦ τ j+k e uv$ where $k=\mathrm{ht}\left(\alpha \right).$

(Khovanov-Lauda)
${ℛ}_{\alpha +\beta }$ is a free (right) ${ℛ}_{\alpha }\otimes {ℛ}_{\beta }$-module with basis $τ σ 1 αβ | σ∈ S k+l S k × S l$ with ${1}_{\alpha \beta }=\sum _{\alpha \in {I}^{\alpha },v\in {I}^{\beta }}{e}_{uv}.$ For $M\in {ℛ}_{\alpha }$-mod and $N\in {ℛ}_{\beta }$-mod, $M-N= Ind ℛ α ⊗ ℛ β ℛ α+β M⊗N .$

${ℛ}_{\alpha }$-mod is the category of finite dimensional $ℤ$-graded ${ℛ}_{\alpha }$-modules: $M= ⊕ i∈ℤ M i with ℛ α j M i ⊆M i+j ,$ where $M= ⊕ i∈ℤ ⊕ u∈ I α M u i with M u i = e u M i .$ The graded character of $M$ is $gch M = ∑ i∈ℤ ∑ u∈ I α dim M u i q i f u .$ Then $gch M·N =gch M ·gch N$ where the right hand side is $⟨,⟩$-shuffle product.

Let $K\left({ℛ}_{\alpha }-\mathrm{mod}\right)$ be the Grothendieck group of ${ℛ}_{\alpha }-\mathrm{mod}.$ $⊕ α∈ Q + K ℛ α -mod → U - M ↦ gch M L g ↦ b g *$ is an algebra isomorphism, where $b g * | g∈G$ is the dual canonical basis of ${U}^{-},$ $L g | g∈G$ are the simple ${ℛ}_{\alpha }$-modules in ${ℛ}_{\alpha }$-mod. $⊕ α∈ Q + K ℛ α -mod → U - M ↦ gch M L g ↢ b g * Δ g ↢ E g *$ where $L\left(g\right),g\in G$ are the simple ${ℛ}_{\alpha }$-modules, if $l\in L$ then $\Delta \left(l\right)=L\left(l\right)$ and $Δ g =Δ l 1 ·…·Δ l k$ if $g={l}_{1}\dots {l}_{k}$ with ${l}_{1},\dots ,{l}_{k}\in G\cap L$ and ${l}_{1}\ge \dots \ge {l}_{k}.$

$\Delta \left(g\right)$ has uniue simple quotient $L\left(g\right).$

## Projective ${ℛ}_{\alpha }$-modules

is the category of finitely generated $ℤ$-generated projective ${ℛ}_{\alpha }$-modules. Let $P\left(i\right)$ be the unique indecomposable projective ${ℛ}_{{\alpha }_{i}}$-module: $P i =span x 1 n e α1 | n∈ ℤ ≥0 ≅ℂ x 1 .$ For $P\in \mathrm{Proj}ℛ\alpha$ and $Q\in \mathrm{Proj}{ℛ}_{\beta }$, define $PQ= Ind ℛ α ⊗ ℛ β ℛ α+β P⊗Q .$ Let $K\left(\mathrm{Proj}{ℛ}_{\alpha }\right)$ be the Grothendieck group of $\mathrm{Proj}{ℛ}_{\alpha }.$

(Khovanov-Lauda, Rouquier)

$U - → ⊕ α∈ Q + K Proj ℛ α P i ↦ P i$ is an algebra isomorphism. Then $b g ↦ P g$ where $\left\{{b}_{g}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in G\right\}$ is the canonical basis of ${U}^{-}$ and $\left\{P\left(g\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in G\right\}$ are the indecomposables in $\mathrm{Proj}{ℛ}_{\alpha }.$