derivation. Let $J$ (resp. $I=J\cup\{0\}$) be a set indexing the vertices of the Dynkin diagram of ${\mathfrak{g}}$ (resp. ${\hat{{\mathfrak{g}}}}$ and ${\hat{{\mathfrak{g}}}}'$). A classical weight is a weight with respect to the algebra ${\mathfrak{g}}$. Denote by $U_q({\mathfrak{g}})$, $U'_q({\hat{{\mathfrak{g}}}})$, and $U_q({\hat{{\mathfrak{g}}}})$ the quantized universal enveloping algebras of ${\mathfrak{g}}$, ${\hat{{\mathfrak{g}}}}'$, and ${\hat{{\mathfrak{g}}}}$ respectively.

There are two main categories of crystals. The first one contains the crystal bases of irreducible integrable $U_q({\hat{{\mathfrak{g}}}})$-modules $V({\Lambda})$ of highest weight ${\Lambda}\in P^+$ where $P^+$ denotes the set of dominant weights of ${\hat{{\mathfrak{g}}}}$. These crystals are infinite-dimensional. The second one contains finite-dimensional crystals corresponding to $U_q'({\hat{{\mathfrak{g}}}})$. Since the set $B$ is finite in this case these crystals are called finite crystals. The level of a finite crystal $B$ is defined as $${\mathrm{lev}}B = \min\{ \langle c,\epsilon(b) \rangle\mid b\in B\}$$ where $\epsilon(b)=\sum_{i\in I} \epsilon_i(b) {\Lambda}_i$ and $\{{\Lambda}_i\mid i\in I\}$ is the ${\mathbb{Z}}$-basis of the weight lattice of ${\hat{{\mathfrak{g}}}}'$. A $U'_q({\hat{{\mathfrak{g}}}})$-crystal can be viewed as a $U_q({\mathfrak{g}})$-crystal by restricting to the $J$-component.

The crystal of each integrable highest weight $U_q({\hat{{\mathfrak{g}}}})$-module can be realized in terms of a semi-infinite tensor product of perfect crystals [@KKMMNN; @KKMMNN2; @KKM]. Perfect crystals are finite crystals with some additional properties (for details see [@KKMMNN Definition 4.6.1]). 