by $$s_i(b) = \begin{cases}
  f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & 
   \text{if $\varphi_i(b)>\epsilon_i(b)$} \\
  b & \text{if $\varphi_i(b)=\epsilon_i(b)$} \\
  e_i^{\epsilon_i(b)-\varphi_i(b)}(b) & 
   \text{if $\varphi_i(b)<\epsilon_i(b)$}.
    \end{cases}$$ It is obvious that $s_i$ is an involution.

Tensor products of crystals
---------------------------

Given two crystals $B$ and $B'$, there is also a crystal obtained by taking the tensor product $B\otimes B'$. As a set $B\otimes B'$ is just given by the Cartesian product of the sets $B$ and $B'$. The weight function ${\mathrm{wt}}$ for $b\otimes b'\in B\otimes B'$ is ${\mathrm{wt}}(b\otimes b')={\mathrm{wt}}(b)+{\mathrm{wt}}(b')$ and the raising and lowering operators $e_i$ and $f_i$ act as follows $$\label{tensor}
\begin{split}
e_i(b\otimes b')&=\begin{cases}
 e_i b \otimes b' & \text{if $\epsilon_i(b)>\phi_i(b')$,}\\
 b \otimes e_i b' & \text{otherwise,}
\end{cases}\\
f_i(b\otimes b')&=\begin{cases}
 f_i b \otimes b' & \text{if $\epsilon_i(b)\ge\phi_i(b')$,}\\
 b \otimes f_i b' & \text{otherwise.}
\end{cases}
\end{split}$$ The reader is warned that this convention is different from Kashiwara’s convention. The order of the tensor factors is interchanged.

Finite and infinite crystals
----------------------------

Let us fix some notation. From now on let ${\mathfrak{g}}$ be a simple complex Lie algebra and ${\hat{{\mathfrak{g}}}}$ be the associated untwisted affine algebra. That is, let ${\hat{{\mathfrak{g}}}}'={\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}] \oplus {\mathbb{C}}c$ be the central extension of the loop algebra of ${\mathfrak{g}}$ and ${\hat{{\mathfrak{g}}}}={\hat{{\mathfrak{g}}}}'\oplus{\mathbb{C}}d$ where $d$ is the degree 