crystal graph of an integrable $U_q({\hat{{\mathfrak{g}}}})$-module. Say that $b\in B$ is an extremal vector of weight $\Lambda\in P$ provided that ${\mathrm{wt}}(b)=\Lambda$ and there exists a family of elements $\{b_w\mid w\in {\hat{W}}\}\subset B$ such that

1.  $b_w=b$ for $w=e$.

2.  If ${\langle h_i\,,\,w{\Lambda}\rangle}\ge 0$ then $e_i(b)=0$ and $f_i^{{\langle h_i\,,\,w{\Lambda}\rangle}}(b_w)=b_{r_i w}$.

3.  If ${\langle h_i\,,\,w{\Lambda}\rangle}\le 0$ then $f_i(b)=0$ and $e_i^{{\langle h_i\,,\,w{\Lambda}\rangle}}(b_w)=b_{r_i w}$.

Say that a $U'_q({\hat{{\mathfrak{g}}}})$-crystal $B$ is *simple* if

1.  $B$ is the crystal base of a finite-dimensional integrable $U'_q({\hat{{\mathfrak{g}}}})$-module.

2.  There is a dominant weight $\Lambda$ with respect to the weight lattice of the classical algebra ${\mathfrak{g}}$ such that $B$ has a unique vector (denoted $u(B)$) of weight $\Lambda$, and the weight of any extremal vector of $B$ is contained in $W\Lambda$. Here $W$ is the Weyl group corresponding to the classical algebra ${\mathfrak{g}}$.

\[thm:simple\] [@AK]

1.  Simple crystals are connected as graphs.

2.  The tensor product of simple crystals is simple.

Let $B$ be a simple $U'_q({\hat{{\mathfrak{g}}}})$-crystal, equipped with a function $D=D_B:B\rightarrow{\mathbb{Z}}$, called its intrinsic energy, which is required to be constant on $J$-components and defined up to a global additive constant. Call the pair $(B,D)$ a graded simple $U'_q({\hat{{\mathfrak{g}}}})$-crystal. We normalize the intrinsic energy function by the requirement 