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A categorification of Fock space using $U_{q} ({sl}_{2})$ representations at a root of unity

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

and

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

Last updates: 3 September 2009

Introduction

Definition of the quantum group

Let

A

be an

n \times n

Cartan matrix and let

d_{1}, \dots, d_{n} \in ℤ_{\geq 0}

be minimal such that

diag (d_{1}, \dots, d_{n}) \cdot A

is symmetric,

A = {(〈 α_{i}, α_{j}^{ˇ} 〉)}_{1 \leq i, j \leq n}

, and

d_{i} = \frac{2}{〈 α_{i}, α_{i}^{ˇ} 〉}

We will use the following notational conventions

q_{i} = q^{d_{i}}

{[n]}_{q} = \frac{q^{n} - q^{- n}}{q - q^{- 1}} = q^{n - 1} + q^{n - 3} + \dots + q^{- (n - 3)} + q^{- (n - 1)}

, for

n \in ℤ

${[n]}_{q}! = {[n]}_{q} {[n - 1]}_{q} \dots {[2]}_{q} {[1]}_{q}$ , for $n \in ℤ_{\geq 0}$ , and

${[\begin{matrix} m \\ n \end{matrix}]}_{q} = \frac{{[m]}_{q}{[m - 1]}_{q}\dots{[m - n + 1]}_{q}}{{[n]}_{q}!}$ , for $n \in ℤ_{\geq 0}$ .

The quantum group is the algebra $U$ generated by

E_{1}, \dots, E_{n}

F_{1}, \dots, F_{n}

, and

K_{1}^{\pm 1}, \dots, K_{n}^{\pm 1}

with relations

K_{i}^{- 1} K_{i} = K_{i} K_{i}^{- 1} = 1

K_{i} K_{j} = K_{j} K_{i}

$K_{i} E_{j} K_{i}^{- 1} = q_{i}^{〈 α_{i}, α_{j}^{ˇ} 〉} E_{j}$ , $K_{i} F_{j} K_{i}^{- 1} = q_{i}^{- 〈 α_{i}, α_{j}^{ˇ} 〉} F_{j}$ , $E_{i} F_{j} - F_{j} E_{i} = δ_{ij} \frac{K_{i} - K_{i}^{- 1}}{q_{i} - q_{i}^{- 1}}$ ,

$\sum_{s = 0}^{1 - 〈 α_{i}, α_{j}^{ˇ} 〉} {(- 1)}^{s} E_{i}^{(1 - 〈 α_{i}, α_{j}^{ˇ} 〉 - s)} E_{j} E_{i}^{(s)} = 0$ , for $i \neq j$ ,

$\sum_{s = 0}^{1 - 〈 α_{i}, α_{j}^{ˇ} 〉} {(- 1)}^{s} F_{i}^{(1 - 〈 α_{i}, α_{j}^{ˇ} 〉 - s)} F_{j} F_{i}^{(s)} = 0$ , for $i \neq j$ ,

where

E_{i}^{(r)} = \frac{E_{i}^{r}}{[r]!}

F_{i}^{(r)} = \frac{F_{i}^{r}}{[r]!}

, for

r \in ℤ_{> 0}

Letting

\begin{matrix} [\begin{matrix} K_{i}; c \\ k \end{matrix}] & = & \prod_{j = 1}^{k} \frac{K_{i} q_{i}^{c - (j - 1)} - K_{i}^{- 1} q_{i}^{- (c - (j - i))}}{q_{i}^{j} - q_{i}^{- j}} \\ = & (\frac{K_{i} q_{i}^{c} - K_{i}^{- 1} q_{i}^{- c}}{q_{i}^{1} - q_{i}^{- 1}}) \dots (\frac{K_{i} q_{i}^{c - (k - 1)} - K_{i}^{- 1} q_{i}^{- (c - (k - 1))}}{q_{i}^{k} - q_{i}^{- k}}), \end{matrix}

the following identity is proved first for $r = 1$ , by induction on $s$ , and then by induction on $r$ with a fixed $s$ (see APPENDIX???),

E_{i}^{(r)} F_{i}^{(s)} = \sum_{k = 0}^{min (r, s)} F_{i}^{(s - k)} {[\begin{matrix} K_{i}; 2 k - s - r \\ k \end{matrix}]}_{q_{i}} E_{i}^{(r - k)}

The restricted integral form of $U$ is the $?$ subalgebra $𝕌$ [THE DS FORM OF U IS NOT WORKING] of $U$ generated by

q^{\pm 1}

E_{i}^{(r)}

F_{i}^{(r)}

and

K_{i}^{\pm 1}

, for

1 \leq i \leq n

and

r \in ℤ_{\geq 0}

[NOTATION INCONSISTENT WITH ABOVE FORM].

The algebra $U$ is a Hopf algebra with

Δ (E_{i}) = E_{i} \otimes 1 + K_{i} \otimes E_{i}

Δ (F_{i}) = F_{i} \otimes K_{i}^{- 1} + 1 \otimes F_{i}

Δ (K_{i}) = K_{i} \otimes K_{i}

$S (E_{i}) = - K_{i}^{- 1} E_{i}$ , $S (F_{i}) = {- F}_{i} K_{i}$ , $S (K_{i}) = K_{i}^{- 1}$ ,

$ε (E_{i}) = 0$ , $ε (F_{i}) = 0$ , $ε (K_{i}) = 1$ .

Since $(E_{i} \otimes 1) (K_{i} \otimes E_{i}) = q^{- 2} (K_{i} \otimes E_{i}) (E_{i} \otimes 1)$ , the $q$ -version of the binomial theorem gives

Δ (E_{i}^{(r)}) = \sum_{k = 0}^{r} q_{i}^{- k (r - k)} E_{i}^{(k)} \otimes K_{i}^{k} E_{i}^{(r - k)}

Δ (F_{i}^{(r)}) = \sum_{k = 0}^{r} q_{i}^{k (r - k)} F_{i}^{(k)} \otimes F_{i}^{(r - k)} K_{i}^{- k}

$S (E_{i}^{(r)}) = {(- 1)}^{r} q_{i}^{r (r - 1)} K_{i}^{- r} E_{i}^{(r)}$ , $S (F_{i}^{(r)}) = {(- 1)}^{r} q_{i}^{- r (r - 1)} F_{i}^{(r)} K_{i}^{r}$ ,

$ε (E_{i}^{(r)}) = 0$ , $ε (F_{i}^{(r)}) = 0$ , $Δ (K_{i}) = K_{i} \otimes K_{i}$ , $S (K_{i}) = K_{i}^{- 1}$ , $ε (K_{i}) = 1$ ,

and hence the subalgebra $ℍ$ [SEE ABOVE] is a Hopf algebra of $U$ . BETTER CHECK THESE!

PUT THE REST OF THE RELATIONS ?? TO MAKE A PRESENTATION OF THE INTEGRAL FORM?? A REF TO CHARI AND PRESSLEY?

The subalgebra $ℍ$ [SEE ABOVE] also has a triangular decomposition

ℍ = ℍ^{-} ℍ_{0} ℍ^{+}

where

ℍ^{+}

is the subalgebra of

U

generated by

E_{i}^{(r)}

$ℍ_{0}$ is the subalgebra of $U$ generated by $K_{i}^{\pm 1}$ and $[\begin{matrix} K_{i}; 0 \\ j \end{matrix}]$ , and

$ℍ^{-}$ is the subalgebra of $U$ generated by $F_{i}^{(r)}$ .

A categorification of Fock space using U q ( sl 2 ) representations at a root of unity

Introduction

Definition of the quantum group

A categorification of Fock space using $U_{q} ({sl}_{2})$ representations at a root of unity