also write $\lambda^{\inv} = \inv(\lambda)$, for $\lambda \in \Lambda$. The involution $\inv$ preserves the bilinear form $(\cdot,\cdot)$ on the weight lattice $\Lambda$ and induces an automorphism on the simple system $\Pi$ such that $ \alpha^{\inv}_i = \alpha_{-i}$ for all $i \in \I$.

Let $\Lambda^{\inv} = \{\mu \in \Lambda \mid \mu^{\inv} +\mu\}$ and $\Lambda_{\inv} = \Lambda/\Lambda^{\inv}$. For $\mu \in \Lambda$, denote by $\ov{\mu}$ the image of $\mu$ under the quotient map. There is a well-defined bilinear pairing $\Z[\alpha_i - \alpha_{-i}]_{i \in \I} \times \BLambda  \rightarrow \Z$, such that $(\sum_{i > 0}a_i(\alpha_i-\alpha_{-i}), \ov{\mu}) :=   \sum_{i > 0}a_i(\alpha_i-\alpha_{-i}, \mu)$ for any $\ov{\mu} \in \BLambda$ with any preimage $\mu\in \Lambda$.

The quantum group 
------------------

The quantum group $\U = \U_q(\mf{sl}_{k+1})$ is defined to be the associative $\Q(q)$-algebra generated by $E_{\alpha_i}$, $F_{\alpha_i}$, $K_{\alpha_i}$, $K^{-1}_{\alpha_i}$, $i \in \I$, subject to the following relations (for $i$, $j \in \I$): $$\begin{aligned}
 K_{\alpha_i} K_{\alpha_i}^{-1} &= K_{\alpha_i}^{-1} K_{\alpha_i} =1,
  \\
 K_{\alpha_i} K_{\alpha_j} &= K_{\alpha_j} K_{\alpha_i},
  \\
 K_{\alpha_i} E_{\alpha_j} K_{\alpha_i}^{-1} &= q^{(\alpha_i, \alpha_j)} E_{\alpha_j}, \\\displaybreak[0]
  K_{\alpha_i} F_{\alpha_j} K_{\alpha_i}^{-1} &= q^{-(\alpha_i, \alpha_j)} F_{\alpha_j}, \\\displaybreak[0]
  E_{\alpha_i} F_{\alpha_j} -F_{\alpha_j} E_{\alpha_i} &= \delta_{i,j} \frac{K_{\alpha_i}
 -K^{-1}_{\alpha_i}}{q-q^{-1}}, \\ \displaybreak[0]
  E_{\alpha_i}^2 E_{\alpha_j} +E_{\alpha_j} E_{\alpha_i}^2 
 &= (q+q^{-1}) E_{\alpha_i} E_{\alpha_j} E_{\alpha_i},  \quad &&\text{if } |i-j|=1, \\
  E_{\alpha_i} E_{\alpha_j} &= E_{\alpha_j} 