Data for quiver Hecke algebras

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

Data for quiver Hecke algebras

$ℱ$ is the free algebra generated by $f_{i}, i \in I .$ $Q^{+} = \sum_{i \in I} ℤ_{\geq 0} α_{i} with \deg (f_{i_{1}} \dots f_{i_{d}}) = α_{i_{1}} + \dots + α_{i_{d}} .$ The symmetric group $S_{d} = ⟨σ_{1} \dots σ_{d - 1}⟩$ acts on $I^{d} = \{words of length d\} = \{u \in I^{*} | l (u) = d\}$ (by rearrangements) with orbit decomposition $I^{d} = \underset{α \in Q^{+}, ht (α) = d}{⊔} I^{α} where I^{α} = \{u \in I * | \deg (u) = α\} .$
Fix a symmetric bilinear form $⟨,⟩ : Q^{+} \times Q^{+} \to ℤ$ given by values $⟨α_{i}, α_{j}⟩ \in ℤ$ so that $A = (⟨{α_{i}}^{\lor}, α_{j}⟩) with {α_{i}}^{\lor} = \frac{2 α_{i}}{⟨α_{i}, α_{i}⟩}$ is a Cartan matrix for a symmetrisable Kac-Moody Lie algebra.
$Γ$ is the graph with vertices $I$ and edges $i$ $j$ if $⟨α_{i}, α_{j}⟩ \neq 0 .$ Fix an orientation $ε_{i j}$ where $ε_{i j} = + 1$ if $i$ $j$ and $ε_{i j} = - 1$ if $i$ $j$ and set $Q_{i j} (u, v) = \{\begin{matrix} 0, if i = j \\ 1, if i \neq j and ⟨α_{i}, α_{j}⟩ = 0 \\ ε_{i j} (v^{- ⟨α_{j}^{\lor}, α_{i}⟩} - u^{- ⟨α_{i}^{\lor}, α_{j}⟩}) if i \neq j and ⟨α_{i}^{\lor}, α_{j}⟩ \neq 0 . \end{matrix}$

Quiver Hecke algebras $ℛ_{α}, α \in Q^{+}$

$ℛ_{α}$ is the associative $ℤ$ -graded algebra given by generators $e_{u}, x_{1} e_{u}, \dots, x_{d} e_{u}, τ_{1} e_{u}, \dots, τ_{d - 1} e_{u}, u \in I^{α}$ with degrees $\deg (e_{u}) = 0, \deg (x_{i} e_{u}) = ⟨u_{i}, u_{i}⟩, \deg (τ_{i} e_{u}) = - ⟨u_{i}, u_{i + 1}⟩$ where $u_{i}$ is the $i$ th letter in $u,$ and relations $e_{u} e_{v} = δ_{u v}, \sum_{u \in I^{α}} e_{u} = 1, x_{i} x_{j} = x_{j} x_{i}, x_{i} e_{u} = e_{u} x_{i},$ $τ_{i} e_{u} = e_{σ_{i} (u)} τ_{i}, τ_{i} τ_{j} = τ_{j} τ_{i}, if i \neq i, i \pm 1$ ${τ_{i}}^{2} e_{u} = Q_{u_{i}, u_{i + 1}} (x_{i}, x_{i + 1}) e_{u},$ $(τ_{i + 1} τ_{i} τ_{i + 1} - τ_{i + 1} τ_{i} τ_{i + 1}) e_{u} = \{\begin{array}{rcl} \frac{1}{x_{i + 2} - x_{i}} (Q_{u_{i + 2}, u_{i + 1}} (x_{i + 2}, x_{i + 1}) - Q_{u_{i + 1}, u_{i}} (x_{i + 1}, x_{i})), if u_{i} = u_{i + 2} \\ 0, if u_{i} \neq u_{i + 2}, \end{array}$ $τ_{i} x_{j} e_{u} = \{\begin{array}{rcl} x_{σ_{i} (j)} τ_{i} e_{u} - ε_{i j} e_{u}, & if u_{i} = u_{i + 1}, \\ x_{σ_{i} (j)} τ_{i} e_{u}, & if u_{i} \neq u_{i + 1} . \end{array}$

Note: $x_{i} = \sum_{u \in I^{α}} x_{i} e_{u}, and τ_{i} = \sum_{u \in I^{α}} τ_{i} e_{u} .$

Structure of $ℛ_{α}$

For each $σ \in S_{α}$ fix a reduced word $σ = σ_{i_{1}} \dots σ_{i_{l}}$ and set $τ_{σ} = τ_{i_{1}} \dots τ_{i_{l}} .$

(Khovanov-Lauda Rouquier)
$ℛ_{α}$ has basis $\{x_{1}^{n_{1}} \dots x_{d}^{n_{d}} τ_{σ} e_{u} | u \in I^{α}, σ \in S_{d}, n_{1}, \dots, n_{d} \in ℤ_{\geq 0}\} .$ If $α \in Q^{+}$ and $β \in Q^{+}$ then $I^{α + β} = \underset{σ \in \frac{S_{k + l}}{S_{k} \times S_{l}}}{\cup} σ (I^{α} . I^{β})$ and there is a homomorphism $\begin{array}{rcl} ℛ_{α} \otimes ℛ_{β} & \to & ℛ_{α + β} \\ e_{u} \otimes e_{v} & \mapsto & x_{i} e_{u v} \\ e_{u} \otimes e_{v} & \mapsto & x_{i} e_{u v} \\ e_{u} \otimes x_{j} e_{v} & \mapsto & x_{j + k} e_{u v} \\ τ_{i} e_{u} \otimes e_{v} & \mapsto & τ_{i} e_{u v} \\ e_{u} \otimes τ_{j} e_{v} & \mapsto & τ_{j + k} e_{u v} \end{array}$ where $k = ht (α) .$

(Khovanov-Lauda)
$ℛ_{α + β}$ is a free (right) $ℛ_{α} \otimes ℛ_{β}$ -module with basis $\{τ_{σ} 1_{α β} | σ \in \frac{S_{k + l}}{S_{k} \times S_{l}}\}$ with $1_{α β} = \sum_{α \in I^{α}, v \in I^{β}} e_{u v} .$ For $M \in ℛ_{α}$ -mod and $N \in ℛ_{β}$ -mod, $M - N = {Ind}_{ℛ_{α} \otimes ℛ_{β}}^{ℛ_{α + β}} (M \otimes N) .$

Graded characters

$ℛ_{α}$ -mod is the category of finite dimensional $ℤ$ -graded $ℛ_{α}$ -modules: $M = \underset{i \in ℤ}{\oplus} M [i] with ℛ_{α} [j] M [i] \subseteq M [i + j],$ where $ℛ_{α} [j] = \{elements of degree j in ℛ_{α}\} .$ $M = \oplus_{i \in ℤ} \oplus_{u \in I^{α}} M_{u} [i] with M_{u} [i] = e_{u} M [i] .$ The graded character of $M$ is $gch (M) = \sum_{i \in ℤ} \sum_{u \in I^{α}} \dim (M_{u} [i]) q^{i} f_{u} .$ Then $gch (M \cdot N) = gch (M) \cdot gch (N)$ where the right hand side is $⟨,⟩$ -shuffle product.

Let $K (ℛ_{α} - \mod)$ be the Grothendieck group of $ℛ_{α} - \mod .$ $\begin{array}{rcl} \underset{α \in Q^{+}}{\oplus} K (ℛ_{α} - \mod) & \to & U^{-} \\ M & \mapsto & gch (M) \\ L (g) & \mapsto & b_{g}^{*} \end{array}$ is an algebra isomorphism, where $\{b_{g}^{*} | g \in G\}$ is the dual canonical basis of $U^{-},$ $\{L (g) | g \in G\}$ are the simple $ℛ_{α}$ -modules in $ℛ_{α}$ -mod. $\begin{array}{rcl} \underset{α \in Q^{+}}{\oplus} K (ℛ_{α} -mod) & \to & U^{-} \\ M & \mapsto & gch (M) \\ L (g) & ↢ & b_{g}^{*} \\ Δ (g) & ↢ & E_{g}^{*} \end{array}$ where $L (g), g \in G$ are the simple $ℛ_{α}$ -modules, if $l \in L$ then $Δ (l) = L (l)$ and $Δ (g) = Δ (l_{1}) \cdot \dots \cdot Δ (l_{k})$ if $g = l_{1} \dots l_{k}$ with $l_{1}, \dots, l_{k} \in G \cap L$ and $l_{1} \geq \dots \geq l_{k} .$

$Δ (g)$ has uniue simple quotient $L (g) .$

Projective $ℛ_{α}$ -modules

$Proj ℛ_{α}$ is the category of finitely generated $ℤ$ -generated projective $ℛ_{α}$ -modules. Let $P (i)$ be the unique indecomposable projective $ℛ_{α_{i}}$ -module: $P (i) = span \{x_{1}^{n} e_{α_{1}} | n \in ℤ_{\geq 0}\} ≅ ℂ [x_{1}] .$ For $P \in Proj ℛ α$ and $Q \in Proj ℛ_{β}$ , define $P Q = {Ind}_{ℛ_{α} \otimes ℛ_{β}}^{ℛ_{α + β}} (P \otimes Q) .$ Let $K (Proj ℛ_{α})$ be the Grothendieck group of $Proj ℛ_{α} .$

(Khovanov-Lauda, Rouquier)

\begin{array}{rcl} U^{-} & \to & \underset{α \in Q^{+}}{\oplus} K (Proj ℛ_{α}) \\ P_{i} & \mapsto & P (i) \end{array}

is an algebra isomorphism. Then

\begin{array}{rcl} b_{g} & \mapsto & P (g) \end{array}

where

\{b_{g} | g \in G\}

is the canonical basis of

U^{-}

and

\{P (g) | g \in G\}

are the indecomposables in

Proj ℛ_{α} .

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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