1 \otimes E_{\alpha_i} + E_{\alpha_i} \otimes K^{-1}_{\alpha_i}, \\
 \Delta (F_{\alpha_i}) &= F_{\alpha_i} \otimes 1 +  K_{\alpha_i} \otimes F_{\alpha_i},\\
 \Delta (K_{\alpha_i}) &= K_{\alpha_i} \otimes K_{\alpha_i}.
 \end{split}\end{aligned}$$ There is a unique $\Qq$-algebra homomorphism $\epsilon: \U \rightarrow \Qq$, called counit, such that $\epsilon(E_{\alpha_i}) = 0$, $\epsilon(F_{\alpha_i}) = 0$, and $\epsilon(K_{\alpha_i}) =1$.

Braid group actions and canonical bases {#subsec:CB}
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Let $W := W_{A_{{k}}} = \mf{S}_{{k}+1}$ be the Weyl group of type $A_{{k}}$. Recall [@Lu94] for each $\alpha_i$ and each finite-dimensional $\U$-module $M$, a linear operator $T_{\alpha_i}$ on $M$ is defined by, for $\lambda \in \Lambda$ and $m \in M_\lambda$, $$T_{\alpha_i}(m) = \sum_{a, b, c \ge 0; -a+b-c
=(\lambda, \alpha_i)}(-1)^b q^{b-ac}E^{(a)}_{\alpha_i}F^{(b)}_{\alpha_i}E^{(c)}_{\alpha_i} m.$$ These $T_{\alpha_i}$’s induce automorphisms of $\U$, denoted by $T_{\alpha_i}$ as well, such that $$T_{\alpha_i}(um) = T_{\alpha_i}(u)T_{\alpha_i}(m), \qquad \text{ for all } u \in \U, m \in M.$$ As automorphisms on $\U$ and as $\Qq$-linear isomorphisms on $M$, the $T_{\alpha_i}$’s satisfy the braid group relations ([@Lu94 Theorem 39.4.3]): $$\begin{aligned}
T_{\alpha_i}T_{\alpha_j} &= T_{\alpha_j}T_{\alpha_i}, &\text{ if } |i-j| >1 ,\\
T_{\alpha_i}T_{\alpha_j}T_{\alpha_i} &= T_{\alpha_j}T_{\alpha_i}T_{\alpha_j}, &\text{ if } |i-j| =1,\end{aligned}$$ Hence for each $w \in W$, $T_w$ can be defined independent of the choices of reduced expressions of $w$. (The $T_{\alpha_i}$ here is $T''_{i,+}$ in [@Lu94]).

Denote 