E_{\alpha_i},  \,\qquad\qquad &&\text{if } |i-j|>1, \\
  F_{\alpha_i}^2 F_{\alpha_j} +F_{\alpha_j} F_{\alpha_i}^2 
 &= (q+q^{-1}) F_{\alpha_i} F_{\alpha_j} F_{\alpha_i},  \quad\, &&\text{if } |i-j|=1,\\
  F_{\alpha_i} F_{\alpha_j} &= F_{\alpha_j} F_{\alpha_i},  \qquad\ \qquad &&\text{if } |i-j|>1.\end{aligned}$$

Let $\U^+$, $\U^0$ and $\U^-$ be the $\Qq$-subalgebra of $\U$ generated by $E_{\alpha_i}$, $K^{\pm 1}_{\alpha_i}$, and $F_{\alpha_i}$ respectively, for $i \in \I$. We introduce the divided power $F^{(a)}_{\alpha_i} = F^a_{\alpha_i}/[a]!$, where $a \ge 0$, $[a] = (q^{a}- q^{-a})/(q-q^{-1})$ and $[a]! = [1][2] \cdots [a]$. Let $\mA =\Z[q,q^{-1}]$. Let $_\mA\U^{+}$ be the $\mA$-subalgebra of $\U^{+}$ generated by $E^{(a)}_{\alpha_i}$ for various $a \ge 0$ and $i \in \I$. Similarly let $_\mA\U^{-}$ be the $\mA$-subalgebra of $\U^{-}$ generated by $E^{(a)}_{\alpha_i}$ for various $a \ge 0$ and $i \in \I$.

\[prop:invol\]

1.  There is an involution $\omega$ on the $\Qq$-algebra $\U$ such that $\omega(E_{\alpha_i}) =F_{\alpha_i}$, $\omega(F_{\alpha_i}) =E_{\alpha_i}$, and $\omega(K_{\alpha_i}) = K^{-1}_{\alpha_i}$ for all $i \in \I$.

2.  There is an anti-linear ($q \mapsto q^{-1}$) bar involution of the $\Q$-algebra $\U$ such that $\ov{E}_{\alpha_i}= E_{\alpha_i}$, $\ov{F}_{\alpha_i}=F_{\alpha_i}$, and $\ov{K}_{\alpha_i}=K_{\alpha_i}^{-1}$ for all $i \in \I$.

    (Sometimes we denote the bar involution on $\U$ by $\psi$.)

Recall that $\U$ is a Hopf algebra with a coproduct $$\begin{aligned}
  \label{eq:coprod}
\begin{split}
\Delta:  &\U \longrightarrow \U \otimes \U,
 \\
 \Delta (E_{\alpha_i}) &= 