The quantum groups GL2, SL2 and PGL2

The quantum groups ${G L}_{2},$ ${S L}_{2}$ and ${P G L}_{2}$

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

Last update: 26 July 2013

The quantum groups ${G L}_{2},$ ${S L}_{2}$ and ${P G L}_{2}$

The quantum group $U_{q} ({G L}_{2})$ is the algebra given by generators $E, F, K_{1}^{\pm 1}, K_{2}^{\pm 1}$ and relations

\begin{matrix} K_{1}^{- 1} K_{1} = K_{1} K_{1}^{- 1} = 1, K_{2}^{- 1} K_{2} = K_{2} K_{2}^{- 1} = 1, K_{1} K_{2} = K_{2} K_{1}, & (1.1) \end{matrix}

\begin{matrix} K_{1} E K_{1}^{- 1} = q^{⟨ ε_{1} - ε_{2}, ε_{1}^{\lor} ⟩} E = q E, K_{2} E K_{2}^{- 1} = q^{⟨ ε_{1} - ε_{2}, ε_{2}^{\lor} ⟩} E = q^{- 1} E, & (1.2) \\ K_{1} F K_{1}^{- 1} = q^{- ⟨ ε_{1} - ε_{2}, ε_{1}^{\lor} ⟩} F = q^{- 1} F, K_{2} F K_{2}^{- 1} = q^{- ⟨ ε_{1} - ε_{2}, ε_{2}^{\lor} ⟩} F = q F, & (1.3) \\ E F - F E = \frac{K_{1} K_{2}^{- 1} - K_{1}^{- 1} K_{2}}{q - q^{- 1}} . & (1.4) \end{matrix}

The coproduct is given by

\begin{matrix} Δ (K_{i}^{\pm 1}) = K_{i}^{\pm 1} \otimes K_{i}^{\pm 1}, ε (K_{i}^{\pm 1}) = 1 S (K_{i}^{\pm 1}) = K_{i}^{\mp 1} (for i = 1, 2) \\ Δ (E) = E \otimes 1 + K_{1} K_{2}^{- 1} \otimes E, ε (E) = 0 S (E) = - K_{1}^{- 1} K_{2} E, \\ Δ (F) = F \otimes K_{1}^{- 1} K_{2} + 1 \otimes F, ε (F) = 0 S (F) = - F K_{1} K_{2}^{- 1}, \end{matrix}

The quantum group $U_{q} ({S L}_{2})$ is the subalgebra of $U_{q} ({G L}_{2})$ generated by $E, F$ and $K = K_{1} K_{2}^{- 1} .$ Alternatively, the quantum group $U_{q} ({S L}_{2})$ is the algebra generated by $E, F$ and $K^{\pm 1}$ with relations

K K^{- 1} = K^{- 1} K = 1, K E K^{- 1} = q^{2} E, K F K^{- 1} = q^{- 2} F, E F - F E = \frac{K - K^{- 1}}{q - q^{- 1}} .

The quantum group $U_{q} ({P G L}_{2})$ is $U_{q} ({G L}_{2})$ with the additional relation $K_{1} K_{2} = 1 .$

In summary, there are Hopf algebra homomorphisms

\begin{matrix} U_{q} ({S L}_{2}) & ↪ & U_{q} ({G L}_{2}) & \overset{π}{⟶} & U_{q} ({P G L}_{2}) & = & \frac{U_{q} ({G L}_{2})}{⟨ K_{1} K_{2} - 1 ⟩} \\ E & ⟼ & E & ⟼ & E \\ F & ⟼ & F & ⟼ & F \\ K & ⟼ & K_{1} K_{2}^{- 1} \\ K_{1} K_{2} & ⟼ & 1 \end{matrix}

The short exact sequences

1 ⟶ {S L}_{2} ⟶ {G L}_{2} \overset{det}{⟶} ℂ^{\times} ⟶ 1 and 1 ⟶ ℂ^{\times} ⟶ {G L}_{2} \overset{π}{⟶} {P G L}_{2} ⟶ 1

produce

\begin{matrix} 𝔥_{P G L}^{*} & ↪ & 𝔥_{G L}^{*} & ⟶ & 𝔥_{S L}^{*} = \frac{𝔥_{G L}^{*}}{ℤ (ε_{1} + ε_{2})} \\ k ε_{1} - k ε_{2} & ⟼ & k ε_{1} - k ε_{2} & for k \in ℤ, \\ a ε_{1} + b ε_{2} & ⟼ & [(a - b) ε_{1}] \end{matrix}

Thus

ℂ [𝔥_{P G L}^{*}] = ℂ [Y^{\pm 1}], ℂ [𝔥_{G L}^{*}] = ℂ [Y_{1}^{\pm 1}, Y_{2}^{\pm 1}], ℂ [𝔥_{S L}^{*}] = ℂ [Y_{1}^{\pm 1}],

with

\begin{matrix} ℂ [𝔥_{P G L}^{*}] & ↪ & ℂ [𝔥_{G L}^{*}] & ⟶ & ℂ [𝔥_{S L}^{*}] = \frac{ℂ [𝔥_{G L}^{*}]}{⟨ Y_{2} = Y_{1}^{- 1} ⟩} \\ Y & ⟼ & Y_{1} Y_{2}^{- 1} \\ Y_{1}^{a} Y_{2}^{b} & ⟼ & Y_{1}^{a - b} \end{matrix}

The dominant weights

{(𝔥_{G}^{*})}^{+} = {a ε_{1} + b ε_{2} | a, b \in ℤ with a \geq b}

are the elements of $𝔥_{G}^{*}$ on the positive side of the hyperplane

𝔥^{ε_{1}^{\lor} - ε_{2}^{\lor}} = {a ε_{1} + b ε_{2} | a, b \in ℤ with a - b = 0} .

and the Weyl group $W_{0} = {1, s_{1}}$ has $s_{1}^{2} = 1,$ with $s_{1}$ reflection in the hyperplane $𝔥^{ε_{1}^{\lor} - ε_{2}^{\lor}} .$ Then

Rep ({P G L}_{2}) ≅ ℂ {[𝔥_{P G L}^{*}]}^{W_{0}}, Rep ({G L}_{2}) ≅ ℂ {[𝔥_{G L}^{*}]}^{w_{0}} Rep ({S L}_{2}) ≅ ℂ {[𝔥_{S L}^{*}]}^{w_{0}}

and the representation theories of ${P G L}_{2}, {G L}_{2}$ and ${S L}_{2}$ are related by

π^{*} : Rep ({P G L}_{2}) ↪ Rep ({G L}_{2}) and Rep ({G L}_{2}) = {{det}^{i} | i \in ℤ} \otimes Rep ({S L}_{2})

so that

\begin{matrix} Rep ({P G L}_{2}) & \overset{π^{*}}{⟶} & Rep ({G L}_{2}) & \overset{res}{⟶} & Rep ({S L}_{2}) = \frac{Rep ({G L}_{2})}{⟨ det ⟩} \\ L (k α) & ⟼ & L (k (ε_{1} - ε_{2})) & for k \in ℤ_{\geq 0}, \\ L (a ε_{1} + b ε_{2}) & ⟼ & L ((a - b) ε_{1}) & for a \geq b \end{matrix}

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