Quantum Groups

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

Last update: 26 July 2013

Quantum groups

For a symbol $q$ use the notation

\begin{matrix} \begin{matrix} {[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_{n \geq 0}, \\ {[\binom{m}{n}]}_{q} = \frac{{[m]}_{q} {[m - 1]}_{q} \dots {[m - n + 1]}_{q}}{{[n]}_{q}!}, for m \in ℤ and n \in ℤ_{\geq 0}, \end{matrix} & (1.1) \end{matrix}

Let $G$ be a complex reductive algebraic group, $B$ a Borel subgroup and $T \subseteq B$ a maximal torus of $G .$ Let

𝔥_{ℤ} = Hom (ℂ^{\times}, T) and 𝔥_{ℤ}^{*} = Hom (T, ℂ^{\times}),

and let

{α_{1}, \dots, α_{n}} \subseteq 𝔥_{ℤ}^{*} and {α_{1}^{\lor}, \dots, α_{n}^{\lor}} \subseteq 𝔥_{ℤ}

the simple roots and coroots corresponding the Borel subgroup $B .$ Let $A$ be the corresponding $n \times n$ Cartan matrix and let $d_{1}, \dots, d_{n} \in ℤ_{> 0}$ be minimal such that $diag (d_{1}, \dots, d_{n}) \cdot A$ is symmetric and let $q_{i} = q^{d_{i}},$

A = {(⟨ α_{i}, α_{j}^{\lor} ⟩)}_{1 \leq i, j \leq n} and d_{i} = \frac{2}{⟨ α_{i}, α_{i} ⟩} and q_{i} = q^{d_{i}} .

The quantum group $U_{q} (G)$ is the Hopf algebra over $ℚ (q)$ generated by

E_{i}, F_{i}, 1 \leq i \leq n, and K_{λ}, λ \in 𝔥_{ℤ},

with relations

\begin{matrix} K_{λ} K_{μ} = K_{λ + μ}, K_{λ} E_{j} K_{- λ} = q^{⟨ λ, α_{j} ⟩} E_{j}, K_{λ} F_{j} K_{- λ} = q^{- ⟨ λ, α_{j} ⟩} F_{j}, \\ E_{i} F_{i} - F_{i} E_{i} = δ_{i j} \frac{K_{α_{i}^{\lor}} - K_{α_{i}^{\lor}}^{- 1}}{q_{i} - q_{i}^{- 1}}, \\ \sum_{s = 0}^{1 - ⟨ α_{i}, α_{j}^{\lor} ⟩} {(- 1)}^{s} {[\binom{1 - ⟨ α_{i}, α_{j}^{\lor} ⟩}{s}]}_{q_{i}} E_{i}^{1 - ⟨ α_{i} α_{j}^{\lor} ⟩ - s} E_{j} E_{i}^{s} = 0, for i \neq j, \\ \sum_{s = 0}^{1 - ⟨ α_{i}, α_{j}^{\lor} ⟩} {(- 1)}^{s} {[\binom{1 - ⟨ α_{i}, α_{j}^{\lor} ⟩}{s}]}_{q_{i}} F_{i}^{1 - ⟨ α_{i} α_{j}^{\lor} ⟩ - s} F_{j} F_{i}^{s} = 0, for i \neq j, \end{matrix}

with

\begin{matrix} Δ (E_{i}) = E_{i} \otimes K_{α_{i}^{\lor}} + 1 \otimes E_{i}, Δ (F_{i}) = F_{i} \otimes 1 + K_{α_{i}^{\lor}}^{- 1} \otimes F_{i}, Δ (K_{λ}) = K_{λ} \otimes K_{λ}, \\ S (E_{i}) = - E_{i} K_{α_{i}^{\lor}}^{- 1}, S (F_{i}) = - K_{α_{i}^{\lor}} F_{i}, S (K_{α_{i}^{\lor}}) = K_{α_{i}^{\lor}}^{- 1}, \\ ε (E_{i}) = 0, ε (F_{i}) = 0, ε (K_{λ}) = 1, \end{matrix}

(see [CPr1994, Section 9.1]). There is a triangular decomposition

\begin{matrix} U_{q}^{> 0} is the subalgebra of U_{q} generated by E_{i}, \\ U_{q} = U_{q}^{< 0} U_{q}^{0} U_{q}^{> 0}, & where & U_{q}^{0} is the subalgebra of U_{q} generated by K_{λ}, \\ U_{q}^{< 0} is the subalgebra of U_{q} generated by F_{i}, \end{matrix}

(see [CPr1994 ,Proposition 9.1.3]). For $1 \leq i \leq n$ and $a, r \in ℤ, r \geq 0,$ let

\begin{matrix} \begin{matrix} K_{i} = K_{α_{i}^{\lor}}, {(E_{i})}^{(r)} = \frac{E_{i}^{r}}{{[r]}_{q_{i}}!}, {(F_{i})}^{(r)} = \frac{F_{i}^{r}}{{[r]}_{q_{i}}!}, and let & (1.2) \end{matrix} \\ {[\binom{q_{i}^{c} K_{i}}{k}]}_{q_{i}} = (\frac{q_{i}^{c} K_{i} - q_{i}^{- c} K_{i}^{- 1}}{q_{i} - q_{i}^{- 1}}) (\frac{q_{i}^{c - 1} K_{i} - q_{i}^{- (c - 1)} K_{i}^{- 1}}{q_{i}^{2} - q_{i}^{- 2}}) \dots (\frac{q_{i}^{c - (k - 1)} K_{i} - q_{i}^{- (c - (k - 1))} K_{i}^{- 1}}{q_{i}^{k} - q_{i}^{- k}}), \end{matrix}

for $i \in {1, \dots, m},$ $c \in ℤ$ and $k \in ℤ_{\geq 0} .$ The identity

\begin{matrix} E_{i}^{(r)} F_{i}^{(s)} = \sum_{k = 0}^{min (r, s)} F_{i}^{(s - k)} {[\binom{q_{i}^{- (s - k + r - k)} K_{i}}{k}]}_{q_{i}} E_{i}^{(r - k)} & (1.3) \end{matrix}

is proved first for $r = 1$ by induction on $s,$ and then by induction on $r$ for a fixed $s$ (see Proposition 1.1 and [Lus1993, Corollary 3.1.9]).

Let $𝔸 = ℤ [q, q^{- 1}] .$ The restricted integral form of $U_{q}$ is the $𝔸 -Hopf$ subalgebra $U_{𝔸}$ generated by

{(E_{i})}^{(r)}, {(F_{i})}^{(r)}, and K_{λ}, for 1 \leq i \leq n and r \in ℤ_{\geq 0} and λ \in 𝔥_{ℤ}

(see [CPr1994, Definition-Proposition 9.3.1]). Intersecting the triangular decomposition of $U_{q}$ with $U_{𝔸},$ the $𝔸 -algebra$ $U_{𝔸}$ has a triangular decomposition

\begin{matrix} U_{𝔸}^{> 0} is the subalgebra of U_{𝔸} generated by {(E_{i})}^{(r)}, \\ U_{𝔸} = U_{𝔸}^{< 0} U_{𝔸}^{0} U_{𝔸}^{> 0}, & where & U_{𝔸}^{0} is the subalgebra of U_{𝔸} generated by K_{λ} and {[\binom{K_{i}; 0}{j}]}_{q_{i}}, \\ U_{𝔸}^{< 0} is the subalgebra of U_{𝔸} generated by {(F_{i})}^{(r)}, \end{matrix}

(see [CPr1994, Proposition 9.3.3]).

As discussed in [Lus1990-2, Section 6], $U_{𝔸} (G)$ is an integral form of $U_{q} (G)$ in the sense that

U_{𝔸} (G) \otimes_{𝔸} ℚ (q) = U_{q} (G)

$(q,$ as an indeterminate, is not a root of unity). As in [CPr1994, Section 9.3A],

\begin{matrix} for ε \in ℂ^{\times} define U_{ε} (G) = U_{𝔸} (G) \otimes_{𝔸} ℂ & (1.4) \end{matrix}

where $ℂ$ is given an $𝔸 -algebra$ via the ring homomorphism which maps $q$ to $ε .$ The remarks just before [CPr1994, Proposition 9.3.5] indicate that if $ε$ is not a root of unity then $U_{ε} (G)$ is the associative algebra generated by $E_{i},$ $F_{i}$ and $K_{λ}$ with relations as in the definition of $U_{q} (G)$ except with $q$ replaced by $ε .$

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