set for the vertices of the Dynkin diagram of ${\mathfrak{g}}$, $\{\alpha_i\in P \mid i\in I \}$ the simple roots, and $\{h_i\in P^*=\mathrm{Hom}_{{\mathbb{Z}}}(P,{\mathbb{Z}})\mid i\in I \}$ the simple coroots. Let $U_q({\mathfrak{g}})$ be the quantized universal enveloping algebra of ${\mathfrak{g}}$. A $U_q({\mathfrak{g}})$-crystal is a nonempty set $B$ equipped with maps ${\mathrm{wt}}:B\rightarrow P$ and $e_i,f_i:B\rightarrow B\cup\{\emptyset\}$ for all $i\in I$, satisfying $$\begin{aligned}
&f_i(b)=b' \Leftrightarrow e_i(b')=b
\text{ if $b,b'\in B$} \\
&{\mathrm{wt}}(f_i(b))={\mathrm{wt}}(b)-\alpha_i \text{ if $f_i(b)\in B$} \\
&{\langle h_i\,,\,{\mathrm{wt}}(b)\rangle}=\varphi_i(b)-\epsilon_i(b)\end{aligned}$$ where $\langle \cdot,\cdot\rangle$ is the natural pairing. Here for $b\in B$, $$\label{eq:eps phi def}
\begin{split}
\epsilon_i(b)&= \max\{n\ge0\mid e_i^n(b)\not=\emptyset \} \\
\varphi_i(b) &= \max\{n\ge0\mid f_i^n(b)\not=\emptyset \}.
\end{split}$$ (It is assumed that $\varphi_i(b),\epsilon_i(b)<\infty$ for all $i\in I$ and $b\in B$.)

A $U_q({\mathfrak{g}})$-crystal $B$ can be viewed as a directed edge-colored graph (the crystal graph) whose vertices are the elements of $B$, with a directed edge from $b$ to $b'$ labeled $i\in I$, if and only if $f_i(b)=b'$. The element $b$ is said to be highest weight if $e_i(b)=\emptyset$ for all $i\in I$. Let $K$ be a subset of $I$. Then the $K$-component of the crystal $B$ is the graph obtained by only considering edges colored by $i\in K$.

We also define the crystal reflection operator $s_i:B\rightarrow B$ 